15. Q15f
Save videos to My Cheatsheet for later, for easy studying.
Video Solution
Q1
Q2
Q3
L1
L2
L3
Similar Question 1
<p>A cube is constructed form the three vectors <code class='latex inline'>\vec{a}</code>, <code class='latex inline'>\vec{b}</code>, and <code class='latex inline'>\vec{c}</code>, as shown below.</p><img src="/qimages/670" /><p>Express each of the diagonals <code class='latex inline'>\vec{AG}, \vec{BH}, \vec{CE}</code>, and <code class='latex inline'>\vec{DF}</code> in terms of <code class='latex inline'>\vec{a}, \vec{b}</code>, and <code class='latex inline'>\vec{c}</code>.</p>
Similar Question 2
<p>For the vectors <code class='latex inline'>\vec{OA} = (-3, 4, 12)</code> and <code class='latex inline'>\vec{OB} = (2, 2, -1)</code>, determine the following:</p><p> <code class='latex inline'>|\vec{OA}|</code>, <code class='latex inline'>|\vec{OB}|</code>, and <code class='latex inline'>|\vec{OP}|</code>.</p>
Similar Question 3
<p>Draw a diagram on the appropriate coordinate system for each of the following vectors:</p><p><code class='latex inline'>\vec{OF}= (0, 0, 5)</code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Simplify the following expression.</p><p><code class='latex inline'> \displaystyle -3(\hat{i} - \hat{k}) - (2\hat{i} + \hat{k}) </code></p>
<p>Draw a diagram on the appropriate coordinate system for each of the following vectors:</p><p><code class='latex inline'>\vec{OP}= (4, -2)</code></p>
<p>Given <code class='latex inline'>\vec{x} = 2\vec{i}-\vec{j}</code> and <code class='latex inline'>\vec{y}=-\vec{i}+5\vec{j}</code>, determine</p><p><code class='latex inline'>|\vec{x} + \vec{y}|</code></p>
<p>Draw a set of <code class='latex inline'>x-, y-</code>, and <code class='latex inline'>z-</code>axes and plot the following points:</p><p><code class='latex inline'>A(1, 0, 0)</code></p>
<p>Determine the magnitude of each Cartesian vector.</p><img src="/qimages/1670" />
<p>Write an ordered triple for the vector.</p><p><code class='latex inline'>\vec{CD}</code> with <code class='latex inline'>C(4, 5, 0)</code> and <code class='latex inline'>D(-3, -3, 5)</code></p>
<p>Where is the following general point located? Express your answer with either an axis or plane.</p><p><code class='latex inline'> F(0,y,0) </code></p>
<p>With the use of a diagram, show that the distributive law, <code class='latex inline'>k(\vec{a} + \vec{b}) =k\vec{a} + k\vec{b}</code> , holds where <code class='latex inline'>k <0, k \in \mathbb{R}</code>.</p>
<p>Prove that the vector <code class='latex inline'>\vec{P_1P_2}</code> from point <code class='latex inline'>P_1(x_1, y_1, z_1)</code> to point <code class='latex inline'>P_2(x_2, y_2, z_2)</code> can be expressed as an ordered triple by subtracting the coordinates of <code class='latex inline'>P_1</code> from the coordinates of <code class='latex inline'>P_2</code>. That is, <code class='latex inline'>P_1P_2 = [x_2 -x_1, y_2 - y1_, z_2 - z_1]</code>.</p>
<p>Draw a diagram on the appropriate coordinate system for each of the following vectors:</p><p><code class='latex inline'>\vec{OD}= (-3, 4)</code></p>
<p>If <code class='latex inline'>\vec{u} = (4, -1)</code> and <code class='latex inline'>\vec{v}= (2, 7)</code>, find</p><p><code class='latex inline'>\displaystyle \vec{v} - \vec{u} </code></p>
<p> Name three vectors with their tails at the origin and their heads on the z-axis.</p>
<p>Each of the points <code class='latex inline'>P(x, y, 0), Q(x, 0, z)</code>, and <code class='latex inline'>R(0, y, z)</code> represent general points on three different planes. Name the three planes to which each corresponds. </p>
<p>Draw a diagram on the appropriate coordinate system for each of the following vectors:</p><p> <code class='latex inline'>\vec{OJ}= (-2, -2, 0)</code></p>
<p> Express the following vectors as an algebraic vector in the form <code class='latex inline'>(a,b)</code>. </p><p><code class='latex inline'> \begin{array}{llllll} &\mid\vec{x}\mid =13, \theta =270^{\circ} \\ \end{array} </code></p>
<p>Express each vector in the form <code class='latex inline'>[a,b]</code>.</p><p>a) <code class='latex inline'>\vec{i}+\vec{j}</code></p><p>b) <code class='latex inline'>-4\vec{i}</code></p><p>c) <code class='latex inline'>2\vec{j}</code></p><p>d) <code class='latex inline'>3\vec{i}+8\vec{j}</code></p><p>e) <code class='latex inline'>-5\vec{i}-2\vec{j}</code></p><p>f) <code class='latex inline'>7\vec{i}-4\vec{j}</code></p><p>g) <code class='latex inline'>-8.2\vec{j}</code></p><p>h) <code class='latex inline'>-2.5\vec{i}+3.3\vec{j}</code></p>
<p>Draw the vector <code class='latex inline'>\vec{AB}</code> joining each pair of points. Then, write the vector in the form [x, y, z]</p><p><code class='latex inline'>A(-1, -7, 2), B(-,3 -2, 5)</code></p>
<p>Given the points P(-6, 1), Q(-2, -1) and R-3, 4), find</p><p><code class='latex inline'> \displaystyle \vec{QP} </code></p>
<p>Three vectors, <code class='latex inline'>\vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c)</code>, and <code class='latex inline'>\vec{OC} = (0, b, c)</code>, are given. </p><p>Determine a formula for each of <code class='latex inline'>|\vec{OA}|, |\vec{OB}|</code>, and <code class='latex inline'>|\vec{OC}|</code>.</p>
<p><em>a)</em> Calculate <code class='latex inline'>\mid\vec{a}\mid</code> when <code class='latex inline'>\vec{a}=(2,3,-2)</code>. </p><p><em>b)</em> Find <code class='latex inline'>\frac{1}{\mid\vec{a}\mid}\vec{a}</code>. Is it a unit vector? </p>
<p>Show that if <code class='latex inline'>\vec{OP} = (5, -10, -10)</code>, then <code class='latex inline'>|\vec{OP}|=15</code>.</p>
<p>Write the following vector in simplified form:</p><p><code class='latex inline'>3(\vec{a} -2\vec{b} -5\vec{c}) - 3(2\vec{a} -4\vec{b} + 2\vec{c}) - (\vec{a}-3\vec{b} +3\vec{c})</code></p>
<p> Can the sum of two unit vectors be a unit vector? Explain. Can the difference? </p>
<p>Using the given diagram, show that the following is true.</p><p><code class='latex inline'> \vec{PQ} </code>=</p><p> <code class='latex inline'>=( \vec{RQ} + \vec{SR}) + \vec{TS} + \vec{PT} </code></p><p><code class='latex inline'>=\vec{RQ} +( \vec{SR}) + \vec{TS} ) + \vec{PT}</code></p><p><code class='latex inline'>=\vec{RQ} + \vec{SR}) + (\vec{TS} + \vec{PT})</code></p><img src="/qimages/669" />
<p>Write an ordered triple for the vector.</p><p><code class='latex inline'>\vec{AB}</code> with <code class='latex inline'>A(0, -3, 2)</code> and <code class='latex inline'>B(0, 4, -4)</code></p>
<p>What is the vector represented in the following diagram?</p><img src="/qimages/2586" />
<p>Plot the following points in <code class='latex inline'>\mathbb{R}^3</code>, using a rectangular prism to illustrate each coordinate.</p><p><code class='latex inline'>B(-2, 1, 1)</code></p>
<p>The point <code class='latex inline'>P(-2, 4, -7)</code> is located in <code class='latex inline'>\mathbb{R}^3</code> as shown on the coordinate axes below</p><img src="/qimages/671" /><p>What its he equation of the plane containing the points <code class='latex inline'>B, C, E</code>, and <code class='latex inline'>P</code>?</p>
<p> Given <code class='latex inline'>\vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2)</code>, and <code class='latex inline'>\vec{z} = (-2, 1, 0)</code>, determine a vector equivalent to each of the following:</p><p> <code class='latex inline'>\displaystyle -2\vec{x} - 3\vec{y} +\vec{z}</code></p>
<p>What is the vector represented in the following diagram?</p><img src="/qimages/2583" />
<p><code class='latex inline'>ABCDEFGH</code> is a rectangular prism.</p><p>Write a single vector that is equivalent to <code class='latex inline'>\vec{EG} + \vec{GH} + \vec{HD} + \vec{DC}</code></p>
<p>Parallelogram OPQR is such that <code class='latex inline'>\vec{OP} = (-7, 24)</code> and <code class='latex inline'>\vec{OR} = (-8, -1)</code>. </p><p>Determine the angle between the diagonals <code class='latex inline'>\vec{OQ}</code> and <code class='latex inline'>\vec{RP}</code>.</p>
<p>Determine the value(s) of <code class='latex inline'>k</code> such that <code class='latex inline'>\vec{u}</code> and <code class='latex inline'>\vec{v}</code> are orthogonal.</p><p> <code class='latex inline'>\vec{u}=[4,1,3], \vec{v}=[-1,5,k]</code></p>
<p> Given <code class='latex inline'>\vec{p} = 2\vec{i} - \vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p><code class='latex inline'>\displaystyle 2\vec{p} - 5\vec{q}</code></p>
<p>Name the vector associated with each point in question 10, express it in component form, and show the vectors associated with each of the points in the diagrams.</p><p> <code class='latex inline'>F(1, -1, -1)</code></p>
<p>Calculate <code class='latex inline'>|(-3, b)|</code> after finding <code class='latex inline'>b</code>.</p>
<p>Given <code class='latex inline'>2\vec{x} + \vec{y} -2\vec{z} = \vec{0}, \vec{x} = (-1, b, c), \vec{y} = (a, -2, c)</code>, and <code class='latex inline'>\vec{z} = (-a, 6 ,c)</code>, determine the value of the unknowns.</p>
<p>In <code class='latex inline'>\mathbb{R}^3</code> , is it possible to locate the point <code class='latex inline'>P(\displaystyle{\frac{1}{2}}, \sqrt{-1}, 3)</code>? Explain.</p>
<p>If <code class='latex inline'>M, N</code>, and <code class='latex inline'>Q</code> are collinear then which one of the following is true?</p><p><strong>A</strong> <code class='latex inline'> \displaystyle \vec{OM} = \frac{7}{8}\vec{ON} + \frac{1}{8}\vec{OQ} </code></p><p><strong>B</strong> <code class='latex inline'> \displaystyle \vec{OM} = \frac{7}{8}\vec{ON} + \frac{2}{8}\vec{OQ} </code></p><p><strong>C</strong> <code class='latex inline'> \displaystyle \vec{OM} = \frac{9}{8}\vec{ON} + \frac{1}{8}\vec{OQ} </code></p><p><strong>D</strong> <code class='latex inline'> \displaystyle 2\vec{OM} = \frac{7}{8}\vec{ON} + \frac{1}{8}\vec{OQ} </code></p>
<p>Are the vectors <code class='latex inline'>\vec{u}=[6,-2,-5]</code> and <code class='latex inline'>\vec{v}=[-12,4,10]</code> collinear? Explain.</p>
<p>A ship’s course is set at a heading of 192°, with a speed of 30 knots. A current is flowing from a bearing of 112°, at 14 knots. Use Cartesian vectors to determine the resultant velocity of the ship.</p>
<p> Express the following vectors as a geometric vector by stating its magnitude and direction. </p><p><code class='latex inline'> \displaystyle \vec{w}=(4,3) </code></p>
<p>Find the exact magnitude of each position vector </p><p>a) [-1, 5, -2] </p><p>b) [3, 3, 3] </p><p>c) [-2, 0, -4]</p>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c}= [1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>7\vec{a}</code></p>
<p>Draw a set of x-, y-, and z-axes and plot the following points:</p><p><code>C(0, 0, -3)</code></p>
<p> Find the components of the unit vector in the direction opposite to <code class='latex inline'>\vec{PQ}</code>, where <code class='latex inline'>\vec{OP} = (11, 19)</code> and <code class='latex inline'>\vec{OQ} = (2, -21)</code>. </p>
<p>Draw the vector <code class='latex inline'>\vec{OA}</code> on a graph, where point <code class='latex inline'>A</code> has coordinates <code class='latex inline'>(6, 10)</code>.</p><p>Draw the vectors <code class='latex inline'>m\vec{OA}</code>, where <code class='latex inline'>m =\displaystyle{\frac{1}{2}}, -\displaystyle{\frac{1}{2}}, 2</code>, and <code class='latex inline'>-2</code>.</p>
<p>Plot the following points in <code class='latex inline'>\mathbb{R}^3</code>, using a rectangular prism to illustrate each coordinate.</p><p> <code class='latex inline'>E(1, -1, 1)</code></p>
<p>Suppose that <code class='latex inline'>\vec{OP}= (a, -3, c)</code> and <code class='latex inline'>\vec{OP}= (-4, b, -8)</code>. What are the corresponding values for <code class='latex inline'>a, b</code>, and <code class='latex inline'>c</code>? Why are we able to be certain that the determined values are correct?</p>
<p>If <code class='latex inline'>\vec{m} = 2\vec{i} -\vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p><code class='latex inline'>|\vec{m}-\vec{n}|</code></p>
<p>Plot the following points in <code class='latex inline'>\mathbb{R}^3</code>, using a rectangular prism to illustrate each coordinate.</p><p><code class='latex inline'>D(1, 1, 1)</code></p>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c}= [1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>\vec{a} + \vec{b} + \vec{c}</code> </p>
<p>On what axis is <code class='latex inline'>A(0, -1, 0)</code> located? Name three other points on this axis. </p>
<p>Determine all unit vectors collinear with [4, 1, -7].</p>
<p>What is the vector represented in the following diagram?</p><img src="/qimages/2585" />
<p>For <code class='latex inline'>A(-1, 3)</code> and <code class='latex inline'>B(2, 5)</code>, draw a coordinate plane and place the points on the graph.</p><p>Determine <code class='latex inline'>|\vec{OA}|</code> and <code class='latex inline'>|\vec{OB}|</code>.</p>
<p>Plot the following points in <code class='latex inline'>\mathbb{R}^3</code>, using a rectangular prism to illustrate each coordinate.</p><p><code class='latex inline'>A(1, 2, 3)</code></p>
<p>In <code class='latex inline'>\mathbb{R}^3</code>, each of the components for each point or vector is a real number. If we use the notation <code class='latex inline'>I^3</code>, where <code class='latex inline'>I</code> represents the set of integers, explain why <code class='latex inline'>\vec{OP} = (-2, 4, -\sqrt{3})</code> would not be an acceptable vector in <code class='latex inline'>\mathbb{I}^3</code>. Why is <code class='latex inline'>\vec{OP}</code> an acceptable vector in <code class='latex inline'>\mathbb{R}^3</code>?</p>
<p> Given <code class='latex inline'>\vec{p} = 2\vec{i} - \vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p><code class='latex inline'>\displaystyle \vec{p} - \vec{q}</code></p>
<p>Express each vector in the form <code class='latex inline'>[a, b, c]</code>. </p><p>a) <code class='latex inline'>3\vec{i}+8\vec{j}</code></p><p>b) <code class='latex inline'>-5\vec{i}-2\vec{k}</code></p><p>c) <code class='latex inline'>7\vec{i}-4\vec{j}+9\vec{k}</code></p>
<p>Prove that <code class='latex inline'>\vec{u}+\vec{v}=[u_1 + v_1, u_2 + v_2, u_3 + v_3]</code> for any two vectors <code class='latex inline'>\vec{u} = [u_1, u_2, u_3]</code> and <code class='latex inline'>\vec{v} = [v_1, v_2, v_3]</code>. </p>
<p><code class='latex inline'>P(2, a-c, a)</code> and <code class='latex inline'>Q(2, 6, 11)</code> represent the same point in <code class='latex inline'>\mathbb{R}^3</code>.</p><p><strong>a)</strong> What are the values of <code class='latex inline'>a</code> and <code class='latex inline'>c</code>?</p><p><strong>b)</strong> Does <code class='latex inline'>|\vec{OP}| = |\vec{OQ}|</code>? Explain.</p>
<p> Given <code class='latex inline'>\vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2)</code>, and <code class='latex inline'>\vec{z} = (-2, 1, 0)</code>, determine a vector equivalent to each of the following:</p><p><code class='latex inline'>\displaystyle \frac{1}{2}\vec{x} - \vec{y} + 3\vec{z}</code></p>
<p>Any unit vector in two dimensions can be written as </p><p><strong>A.</strong> <code class='latex inline'>(\cos\theta, \sin\theta)</code> where <code class='latex inline'>\theta</code>is the angle between the vector and the <code class='latex inline'>x-</code>axis. </p><p><strong>B.</strong> <code class='latex inline'>(\sin\theta, \cos\theta)</code> where <code class='latex inline'>\theta</code>is the angle between the vector and the <code class='latex inline'>x-</code>axis. </p><p><strong>C</strong> <code class='latex inline'>(\cos\theta, \tan\theta)</code> where <code class='latex inline'>\theta</code>is the angle between the vector and the <code class='latex inline'>x-</code>axis. </p><p><strong>D</strong> <code class='latex inline'>(\sin\theta, \tan\theta)</code> where <code class='latex inline'>\theta</code>is the angle between the vector and the <code class='latex inline'>x-</code>axis. </p>
<p>For the vectors <code class='latex inline'>\vec{OA} = (-3, 4, 12)</code> and <code class='latex inline'>\vec{OB} = (2, 2, -1)</code>, determine the following:</p><p><code class='latex inline'>\vec{AB}</code> and <code class='latex inline'>|\vec{AB}|</code>. What does <code class='latex inline'>\vec{AB}</code> represent?</p>
<p>If <code class='latex inline'>\vec{a} = 3\vec{i} -4\vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{b} = -2\vec{i} + 3\vec{j}-\vec{k}</code>, express each of the following in terms of <code class='latex inline'>\vec{i}, \vec{j}</code>, and <code class='latex inline'>\vec{k}.</code></p><p><code class='latex inline'>2\vec{a} -3\vec{b}</code></p>
<p>Draw a diagram on the appropriate coordinate system for each of the following vectors:</p><p><code class='latex inline'>\vec{OC}= (2, 4, 5)</code></p>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c} =[1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>\vec{b} + \vec{c} + \vec{a}</code> </p>
<p><code class='latex inline'>\triangle ABC</code> has vertices at <code class='latex inline'>A(2, 3)</code>, <code class='latex inline'>B(6, 6)</code> and <code class='latex inline'>C(-4, 11)</code>.</p><p>Verify that triangle ABC is a right triangle.</p>
<p>Parallelogram OPQR is such that <code class='latex inline'>\vec{OP} = (-7, 24)</code> and <code class='latex inline'>\vec{OR} = (-8, -1)</code>. </p><p>Determine the angle between the vectors <code class='latex inline'>\vec{OR}</code> and <code class='latex inline'>\vec{OP}</code>.</p>
<p>Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following: </p><p> <code class='latex inline'>|\vec{OA}|</code></p>
<p>Draw vectors <code class='latex inline'>\vec{u}</code> and <code class='latex inline'>\vec{v}</code> at right angles to each other with <code class='latex inline'>|\vec{u}|=3cm</code> and <code class='latex inline'>|\vec{v}|=4.5cm</code>. Then, draw the following linear combinations of <code class='latex inline'>\vec{u}</code> and <code class='latex inline'>\vec{v} </code>.</p><p>a) <code class='latex inline'>\vec{u}+\vec{v}</code></p><p>b) <code class='latex inline'>2\vec{u}-\vec{v}</code></p><p>c) <code class='latex inline'>0.5\vec{u}+2\vec{v}</code></p><p>d) <code class='latex inline'>\vec{v}-\vec{u}</code></p>
<p>If <code class='latex inline'>\vec{u} = (4, -1)</code> and <code class='latex inline'>\vec{v}= (2, 7)</code>, find</p><p><code class='latex inline'>\displaystyle -8 \vec{u} </code></p>
<p>For <code class='latex inline'>A(-1, 3)</code> and <code class='latex inline'>B(2, 5)</code>, draw a coordinate plane and place the points on the graph.</p> <ul> <li>Draw vectors <code class='latex inline'>\vec{AB}</code> and <code class='latex inline'>\vec{BA}</code>, and give vectors in component form equivalent to each of these vectors.</li> </ul>
<p> Given <code class='latex inline'>\vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2)</code>, and <code class='latex inline'>\vec{z} = (-2, 1, 0)</code>, determine a vector equivalent to each of the following:</p><p><code class='latex inline'>\displaystyle \vec{x} -2\vec{y} - \vec{z}</code></p>
<p>If <code class='latex inline'>\vec{a} = 3\vec{i} -4\vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{b} = -2\vec{i} + 3\vec{j}-\vec{k}</code>, express each of the following in terms of <code class='latex inline'>\vec{i}, \vec{j}</code>, and <code class='latex inline'>\vec{k}.</code></p><p> <code class='latex inline'>\vec{a} + 5\vec{b}</code></p>
<p>For the point following, draw the x-axis, y-axis, and z-axis and accurately draw the position vectors. </p><p><code class='latex inline'> \displaystyle P(2,3,-7) </code></p>
<p> Describe the form of the coordinates of all points that are equidistant from the x—, y- and z-axes</p>
<p>Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following: </p><p><code class='latex inline'>|\vec{OB}|</code></p>
<p>The point <code class='latex inline'>P(-2, 4, -7)</code> is located in <code class='latex inline'>\mathbb{R}^3</code> as shown on the coordinate axes below</p><img src="/qimages/671" /><p><strong>a)</strong> Determine the coordinates of points <code class='latex inline'>A,B,C,D,E</code>, and <code class='latex inline'>F</code>.</p><p><strong>b)</strong> What are the vectors associated with each of the points in part a.?</p>
<p> Given <code class='latex inline'>\vec{p} = 2\vec{i} - \vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p><code class='latex inline'>\displaystyle -2\vec{p} +5\vec{q}</code></p>
<p>Determine the value of <code class='latex inline'>x</code> and <code class='latex inline'>y</code> in each of the following:</p><p><code class='latex inline'>-2(x, x + y) - 3(6, y) = (6, 4)</code></p>
<p>If * is an operation on a set, <code class='latex inline'>S</code>, the element <code class='latex inline'>x</code>, such that <code class='latex inline'>a*x =a</code>, is called the identity element for the operation *.</p><p><strong>a)</strong> For the addition of numbers, what is the identity element?</p><p><strong>b)</strong> For the multiplication of numbers, what is the identity element?</p><p><strong>c)</strong> For the addition of vectors, what is the identity element?</p><p><strong>d)</strong> For scalar multiplication, what is the identity element?</p>
<p>Write the force as a Cartesian vector.</p><p>25 N applied downward</p>
<p>Determine the angle between the vectors <code class='latex inline'>\vec{g} =[6 ,1, 2]</code> and <code class='latex inline'>\vec{h} = [-5, 3, 6]</code>.</p>
<p>Prove thta the magnitude of a vector can be equal zero if and only if the vector is the zero vector, <code class='latex inline'>\vec{0}</code>.</p>
<p><strong>a)</strong> Write the vector <code class='latex inline'>\vec{OA} = (-1, 2, 4)</code> using the standard unit vectors.</p><p><strong>b)</strong> Determine <code class='latex inline'>|\vec{OA}|</code>.</p>
<p>Find a single vector equivalent to each of the following:</p><p><code class='latex inline'>-3(4, -9) - 9(2, 3)</code></p>
<p>If <code class='latex inline'>\vec{u} = (4, -1)</code> and <code class='latex inline'>\vec{v}= (2, 7)</code>, find</p><p><code class='latex inline'> \displaystyle 8\vec{u} </code></p>
<p>If <code class='latex inline'>\vec{a} = 3\vec{i} -4\vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{b} = -2\vec{i} + 3\vec{j}-\vec{k}</code>, express each of the following in terms of <code class='latex inline'>\vec{i}, \vec{j}</code>, and <code class='latex inline'>\vec{k}.</code></p><p><code class='latex inline'>2(\vec{a} -\vec{b}) -3(-2\vec{a} - 7\vec{b})</code></p>
<p>Are the vectors </p><p><code class='latex inline'>\vec{a} = (0, 0, 1), \vec{b} = (0, 0, 2), \vec{b} = (0, 0, 101)</code></p><p> collinear? Explain.</p>
<p>If <code class='latex inline'>a\vec{i} + 5\vec{j} = (-3, b)</code>, determine the values of <code class='latex inline'>a</code> and <code class='latex inline'>b</code>.</p>
<p> If <code class='latex inline'>\vec{a} = 3\vec{i} + 2\vec{j} - \vec{k}</code> and <code class='latex inline'>\vec{b} = -2\vec{i} + \vec{j}</code>, calculate each magnitude.</p><p><code class='latex inline'> \displaystyle |2 \vec{a} - 3\vec{b}| </code></p>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c} =[1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>(\vec{a} + \vec{b}) \cdot (\vec{a}- \vec{b})</code> </p>
<p>Determine two vectors that are orthogonal to, the vector.</p><p><code class='latex inline'>\vec{e} = [-4, -9, 3]</code></p>
<p>Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following: </p><p><code class='latex inline'>\vec{AB}</code></p>
<p>Three vectors, <code class='latex inline'>\vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c)</code>, and <code class='latex inline'>\vec{OC} = (0, b, c)</code>, are given. </p><p>Determine <code class='latex inline'>\vec{AB}</code>. What does <code class='latex inline'>\vec{AB}</code> represent?</p>
<p>Let <code class='latex inline'>\vec{a} = (4, 7)</code> and <code class='latex inline'>\vec{b} =(2, -9)</code>.</p><p>a) Plot the two vectors.</p><p>b) Which is greater, <code class='latex inline'>|\vec{a} + \vec{b}|</code> or <code class='latex inline'>|\vec{a}| + |\vec{b}|</code>?</p><p>c) Will this be true for all pairs of vectors? Justify your answer with examples.</p>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c} =[1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>\vec{a}\cdot \vec{c}</code> </p>
<p>Find a single vector equivalent to each of the following:</p><p><code class='latex inline'>-\displaystyle{\frac{1}{2}}(6, -2) \displaystyle{\frac{2}{3}}(6, 15)</code></p>
<p>Write the force as a Cartesian vector.</p><p>230 N applied to the east</p>
<p>Determine the value(s) of <code class='latex inline'>k</code> such that <code class='latex inline'>\vec{u}</code> and <code class='latex inline'>\vec{v}</code> are orthogonal.</p><p><code class='latex inline'>\vec{u}=[11,3,2k], \vec{v}=[k,4,k]</code></p>
<p>Redraw the following three vectors and illustrate the associate law.</p><img src="/qimages/668" />
<p>Given <code class='latex inline'>\vec{x} = 2\vec{i}-\vec{j}</code> and <code class='latex inline'>\vec{y}=-\vec{i}+5\vec{j}</code> , find a vector equivalent to each of the following:</p><p><code class='latex inline'>-(\vec{x} + 2\vec{y}) + 3(-\vec{x} -3\vec{y})</code></p>
<p>Draw a diagram on the appropriate coordinate system for each of the following vectors:</p><p><code class='latex inline'>\vec{OF}= (0, 0, 5)</code></p>
<p> Express the following vectors as an algebraic vector in the form <code class='latex inline'>(a,b)</code>. </p><p><code class='latex inline'> \begin{array}{llllll} & \mid \vec{v}\mid =36, \theta =330^{\circ} \end{array} </code></p>
<p>Write the vector <code class='latex inline'>\vec{OB} = 3\vec{i} + 4\vec{j} - 4\vec{k}</code> in component form and calculate its magnitude.</p>
<p>If <code class='latex inline'>\vec{m} = 2\vec{i} -\vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p><code class='latex inline'>2\vec{m} + 3\vec{n}</code></p>
<p>Given <code class='latex inline'>\vec{x} = 2\vec{i}-\vec{j}</code> and <code class='latex inline'>\vec{y}=-\vec{i}+5\vec{j}</code> , find a vector equivalent to each of the following:</p><p><code class='latex inline'>2(\vec{x} + 3\vec{y}) -3(\vec{y} + 5\vec{x})</code></p>
<p>Write the force as a Cartesian vector.</p><p>650 N applied to the west</p>
<p>For the point following, draw the x-axis, y-axis, and z-axis and accurately draw the position vectors. </p><p><code class='latex inline'> \displaystyle N(-3,5,3) </code></p>
<p>The terminal point of vector <code class='latex inline'>\vec{DE} = [-4, 2, 6]</code> is <code class='latex inline'>E(3, 3, 1)</code>. Determine the coordinates of the initial point, D.</p>
<p>If <code class='latex inline'>\vec{m} = 2\vec{i} -\vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p> <code class='latex inline'>-5\vec{m}</code></p>
<p>Draw a set of x-, y-, and z-axes and plot the following points:</p><p><code>F(0, 2, 3)</code></p>
<p>Draw a set of <code class='latex inline'>x-, y-</code>, and <code class='latex inline'>z-</code>axes and plot the following points:</p><p><code class='latex inline'>D(2, 3, 0)</code></p>
<p>If <code class='latex inline'>\vec{m} = 2\vec{i} -\vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p> <code class='latex inline'>|\vec{m}+\vec{n}|</code></p>
<p><code class='latex inline'>A(5, 0)</code> and <code class='latex inline'>B(0, 2)</code> are points on the x- and y-axes, respectively. </p><p>Find the coordinates of point <code class='latex inline'>P(a, 0)</code> on the x-axis such that <code class='latex inline'>|\vec{PA}| = |\vec{PB}|</code>.</p>
<p>The direction angles of a vector are all equal. The vector is a three-dimensional vector. Find the direction angles to the nearest degree. </p><p><a href="https://youtu.be/860hCOgNmJo">HINT</a></p>
<p>For the point following, draw the x-axis, y-axis, and z-axis and accurately draw the position vectors. </p><p><code class='latex inline'> \displaystyle Q(-4,-9,5) </code></p>
<p>Draw a diagram illustrating the set of points</p><p><code class='latex inline'>\{(x, y, z)\in \mathbb{R}^3 \vert 0 \leq x \leq 1, 0\leq y \leq 1, 0\leq z \leq 1\}</code>.</p>
<p>Determine the equation of the plane containing the points <code class='latex inline'>A, B</code>, and <code class='latex inline'>C</code>.</p>
<p>Draw a diagram on the appropriate coordinate system for each of the following vectors:</p><p><code class='latex inline'>\vec{OM}= (-1, 3, -2)</code></p>
<p>For <code class='latex inline'>A(-1, 3)</code> and <code class='latex inline'>B(2, 5)</code>, draw a coordinate plane and place the points on the graph.</p> <ul> <li>Calculate <code class='latex inline'>|\vec{AB}|</code> and state the value of <code class='latex inline'>|\vec{BA}|</code>.</li> </ul>
<p> Given <code class='latex inline'>\vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2)</code>, and <code class='latex inline'>\vec{z} = (-2, 1, 0)</code>, determine a vector equivalent to each of the following:</p><p><code class='latex inline'>\displaystyle 3\vec{x} + 5\vec{y} + 3\vec{z}</code></p>
<p>The point <code class='latex inline'>P(-2, 4, -7)</code> is located in <code class='latex inline'>\mathbb{R}^3</code> as shown on the coordinate axes below</p><img src="/qimages/671" /><p>How far below the xy-plane is the rectangle <code class='latex inline'>DEPF</code>?</p>
<p>Find a and b such that <code class='latex inline'>\vec{u}</code> and <code class='latex inline'>\vec{v}</code> are collinear.</p><p><code class='latex inline'>\vec{u} = a\vec{i} + 2\vec{j}</code>, <code class='latex inline'>\vec{v} = 3\vec{i} - 6\vec{j} - b\vec{k}</code>.</p>
<p>If <code class='latex inline'>\vec{x} = \displaystyle{\frac{2}{3}}\vec{y} + \displaystyle{\frac{1}{3}}\vec{z}, \vec{x} - \vec{y } = \vec{a}</code>, and <code class='latex inline'>\vec{y} - \vec{z} = \vec{b}</code>, show that <code class='latex inline'>\vec{a} = -\displaystyle{\frac{1}{3}}\vec{b}</code>.</p>
<p>For the vectors <code class='latex inline'>\vec{OA} = (-3, 4, 12)</code> and <code class='latex inline'>\vec{OB} = (2, 2, -1)</code>, determine the following:</p><p> the components of vector <code class='latex inline'>\vec{OP}</code>, where <code class='latex inline'>\vec{OP} = \vec{OA} +\vec{ OB}</code></p>
<p>A triangle has vertices at the points A(5, 0, 0), B(0, 5, 0), and C(0, 0, 5).</p><p>a) What type of triangle is <code class='latex inline'>\triangle ABC</code>? Explain.</p><p>b) Identify the point in the interior of the triangle that is closest to the origin.</p>
<p>Given <code class='latex inline'>\vec{x} = 2\vec{i}-\vec{j}</code> and <code class='latex inline'>\vec{y}=-\vec{i}+5\vec{j}</code> , find a vector equivalent to each of the following:</p><p><code class='latex inline'>3\vec{x} - \vec{y}</code></p>
<p>For the vector <code class='latex inline'>\vec{OA}= 3\vec{i} -4\vec{j}</code>, calculate <code class='latex inline'>|\vec{OA}|</code>.</p>
<p>Draw the vector <code class='latex inline'>\vec{AB}</code> joining each pair of points. Then, write the vector in the form [x, y, z]</p><p><code class='latex inline'>A(0, 0, 6), B(0, -5, 0)</code></p>
<p> Identify the type of triangle with vertices A(2, 3, -5), B(-4, 8, 1), and C(6, -4, 0).</p>
<p> Express the following vectors as a geometric vector by stating its magnitude and direction. </p><p><code class='latex inline'> \displaystyle \vec{x}= (0,8) </code></p>
<p><strong>a)</strong> What is the equation of the plane that contains the points <code class='latex inline'>M(1, 0, 3), N(4, 0, 6)</code> and <code class='latex inline'>P(7, 0, 9)</code>? Explain your answer.</p><p><strong>b)</strong> Explain why the plane that contains the points <code class='latex inline'>M,N</code>,and <code class='latex inline'>P</code> also contains the vectors <code class='latex inline'>\vec{OM}</code>, <code class='latex inline'>\vec{ON}</code> and <code class='latex inline'>\vec{OP}</code>.</p>
<p>a) Draw a set of x-, y-, and z-axes and plot the following points: <code class='latex inline'>A(3, 2, -4), B(1, ,1 -4)</code>, and <code class='latex inline'>C(0, 1, -4)</code>.</p>
<p>Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following: </p><p><code class='latex inline'>\vec{BA}</code></p>
<p>Draw the vector <code class='latex inline'>\vec{OA}</code> on a graph, where point <code class='latex inline'>A</code> has coordinates <code class='latex inline'>(6, 10)</code>.</p><p>Which of these vectors have the same magnitude?</p>
<p>Describe the form of the coordinates of all points that are equidistant from the x— and y—axes. </p>
<p>Draw a set of <code class='latex inline'>x-, y-</code>, and <code class='latex inline'>z-</code>axes and plot the following points:</p><p><code class='latex inline'>D(2, 0, 3)</code></p>
<p>Express each vector as a sum of the vectors <code class='latex inline'>\vec{i}, \vec{j}</code>, and <code class='latex inline'>\vec{k}</code>.</p><p> [-3,-6,9]</p>
<p>Describe the coordinates of all points that are 10 units from both the x- and z-axes.</p>
<p>Determine two vectors that are orthogonal to, the vector.</p><p><code class='latex inline'>\vec{e} = [3, -1, 4]</code></p>
<p>Parallelogram OBCA is determined by the vectors <code class='latex inline'>\vec{OA}=(6, 3)</code> and <code class='latex inline'>\vec{OB}=(11, -6)</code>.</p><p><strong>a)</strong> Determine <code class='latex inline'>\vec{OC}, \vec{BA}</code>, and <code class='latex inline'>\vec{BA}</code>.</p><p><strong>b)</strong> Verify that <code class='latex inline'>|\vec{OA}|= |\vec{BA}|</code>.</p>
<p>Simplify the following expression.</p><p><code class='latex inline'> \displaystyle 3(\hat{i} -2\hat{j} + 3\hat{k}) - (-\hat{i} + 4\hat{j} - 3\hat{k}) </code></p>
<p>Three vertices of a parallelogram are <code class='latex inline'>D(2, 1, 3), E(-4, 2, 0)</code>, and <code class='latex inline'>F(6, -2, 4)</code>. Find all possible locations of the fourth vertex. </p>
<p>A cube is constructed form the three vectors <code class='latex inline'>\vec{a}</code>, <code class='latex inline'>\vec{b}</code>, and <code class='latex inline'>\vec{c}</code>, as shown below.</p><img src="/qimages/670" /><p>Express each of the diagonals <code class='latex inline'>\vec{AG}, \vec{BH}, \vec{CE}</code>, and <code class='latex inline'>\vec{DF}</code> in terms of <code class='latex inline'>\vec{a}, \vec{b}</code>, and <code class='latex inline'>\vec{c}</code>.</p>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c} =[1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>\vec{c} - \vec{b} </code> </p>
<p> If <code class='latex inline'>\vec{a} = 3\vec{i} + 2\vec{j} - \vec{k}</code> and <code class='latex inline'>\vec{b} = -2\vec{i} + \vec{j}</code>, calculate each magnitude.</p><p><code class='latex inline'> \displaystyle |\vec{a} -\vec{b}| </code></p>
<p>Rectangle ABCD has vertices at <code class='latex inline'>A(2, 3), B( -6, 9), C(x, y),</code> and <code class='latex inline'>D(8, 11)</code>. </p><p><strong>a)</strong> Draw a sketch of the points A, B, and D, and locate point C on your graph.</p><p><strong>b)</strong> Explain how you can determine the coordinates of point C. </p>
<p>For the point following, draw the x-axis, y-axis, and z-axis and accurately draw the position vectors. </p><p><code class='latex inline'> \displaystyle R(5,-5,-1) </code></p>
<p>Draw a vector diagram to illustrate each combination of vectors </p><p>a) <code class='latex inline'>\vec{u}+\vec{u}+\vec{u}</code></p><p>b) <code class='latex inline'>2\vec{u}-3\vec{v}-3\vec{u}+\vec{v}</code></p><p>c) <code class='latex inline'>3(\vec{u}+\vec{v})-3(\vec{u}-\vec{v})</code></p><p>d) <code class='latex inline'>3\vec{u}+2\vec{v}-2(\vec{v}-\vec{u})+(-3\vec{v})</code></p><p>e) <code class='latex inline'>-(\vec{u}+\vec{v})-4(\vec{u}-2\vec{v})</code></p><p>f) <code class='latex inline'>2(\vec{u}+\vec{v})-2(\vec{u}+\vec{v})</code></p>
<p> Express the following vectors as an algebraic vector in the form <code class='latex inline'>(a,b)</code>. </p><p><code class='latex inline'> \begin{array}{llllll} & \mid \vec{u} \mid = 12, \theta = 135^{\circ} \end{array} </code></p>
<p>Use Cartesian vectors to prove that for any vectors <code class='latex inline'>\vec{a}, \vec{b}</code> and <code class='latex inline'>\vec{c}</code> and scalar <code class='latex inline'>k\in\mathbb{R}</code>,</p><p> <code class='latex inline'>k(\vec{a}+\vec{b})=k\vec{a}+k\vec{b}</code></p>
<p>Locate the points <code class='latex inline'>A(4, -4, -2), B(-4, 4, 2)</code> and <code class='latex inline'>C(4, 4, -2)</code> using coordinate axes that you construct yourself. Draw the corresponding rectangular box (prism) for each, and label the coordinates of its vertices.</p>
<p> Express the following vectors as a geometric vector by stating its magnitude and direction. </p><p><code class='latex inline'> \displaystyle \vec{v} =(-4\sqrt{3}, -12) </code></p>
<p>If <code class='latex inline'>\vec{a} = (1, 3, -3)</code>, <code class='latex inline'>\vec{b} =(-3, 6 ,12)</code>, and <code class='latex inline'>\vec{c} = (0, 8, 1)</code>, determine <code class='latex inline'>|\vec{a} + \displaystyle{\frac{1}{3}}\vec{b} - \vec{c}|</code>.</p>
<p>In the trapezoid <code class='latex inline'>TXYZ</code>, <code class='latex inline'>\vec{TX} = 2\vec{ZY}</code>. If the diagonals meet at <code class='latex inline'>O</code>, find an expression for <code class='latex inline'>\vec{TO}</code> in terms of <code class='latex inline'>\vec{TX}</code> and <code class='latex inline'>\vec{TZ}</code>.</p>
<p>Express each vector as a sum of the vectors <code class='latex inline'>\vec{i}, \vec{j}</code>, and <code class='latex inline'>\vec{k}</code>.</p><p> [5,0,-7]</p>
<p>If <code class='latex inline'>\vec{OP} = (-2, 3, 6)</code> and <code class='latex inline'>B(4, -2, 8)</code>, determine the coordinates of point A such that <code class='latex inline'>\vec{OP} =\vec{AB}</code>.</p>
<p>Given <code class='latex inline'>\vec{x} + \vec{y} = -\vec{i} + 2\vec{j} + 5\vec{k}</code>, and <code class='latex inline'>\vec{n} = -2\vec{i} + \vec{j} + 2\vec{k}</code>, determine <code class='latex inline'>\vec{x}</code> and <code class='latex inline'>\vec{y}</code>.</p>
<p>What is the magnitude of <code class='latex inline'>\displaystyle{\frac{1}{|\vec{v}|}}\vec{v}</code> for any vector <code class='latex inline'>\vec{v}</code>?</p>
<p>Which vector is not collinear with <code class='latex inline'>\vec{a} = (6 -4)</code>?</p><p><strong>A</strong> <code class='latex inline'>\vec{b} = (3, -2)</code></p><p><strong>B</strong> <code class='latex inline'>\vec{c} = (-6, -4)</code></p><p><strong>C</strong> <code class='latex inline'>\vec{d} = (-6, 4)</code></p><p><strong>D</strong> <code class='latex inline'>\vec{e} = (-9, 6)</code></p>
<p>Find a and b such that <code class='latex inline'>\vec{u}</code> and <code class='latex inline'>\vec{v}</code> are collinear.</p><p><code class='latex inline'>\vec{u} = [a, 3, 6]</code>, <code class='latex inline'>\vec{v} = [-8, 12, b]</code></p>
<p>If <code class='latex inline'>2\vec{x} + 3\vec{y} = \vec{a}</code> and <code class='latex inline'>-\vec{x} + 5\vec{y} = 6\vec{b}</code>, express <code class='latex inline'>\vec{x}</code> and <code class='latex inline'>\vec{y}</code> in terms of <code class='latex inline'>\vec{a}</code> and <code class='latex inline'>\vec{b}</code>.</p>
<p>For the vectors <code class='latex inline'>\vec{OA} = (-3, 4, 12)</code> and <code class='latex inline'>\vec{OB} = (2, 2, -1)</code>, determine the following:</p><p> <code class='latex inline'>|\vec{OA}|</code>, <code class='latex inline'>|\vec{OB}|</code>, and <code class='latex inline'>|\vec{OP}|</code>.</p>
<p>A parallelogram has three of its vertices at <code class='latex inline'>A(-1, 2) ,B(7, -2),</code> and <code class='latex inline'>C(2, 8)</code>.</p><p><strong>a)</strong> Draw a grid and locate each of these points.</p><p><strong>b)</strong> On your grid, draw the three locations for a fourth point that would make a parallelogram with points A, B, and C.</p><p><strong>c)</strong> Determine all possible coordinates for the point described in part b.</p>
<p>Plot the following points in <code class='latex inline'>\mathbb{R}^3</code>, using a rectangular prism to illustrate each coordinate.</p><p> <code class='latex inline'>C(1, -2, 1)</code></p>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c} =[1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>\vec{b}\cdot (\vec{a} + \vec{c}) </code> </p>
<p>Three vectors, <code class='latex inline'>\vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c)</code>, and <code class='latex inline'>\vec{OC} = (0, b, c)</code>, are given. </p><p>In a sentence, describe what each vector represents.</p>
<p><code class='latex inline'>\triangle ABC</code> has vertices at <code class='latex inline'>A(2, 3)</code>, <code class='latex inline'>B(6, 6)</code> and <code class='latex inline'>C(-4, 11)</code>.</p><p><strong>a)</strong> Sketch and label each of the points on a graph.</p><p><strong>b)</strong> Calculate teach of the lengths <code class='latex inline'>|\vec{AB}|</code>, <code class='latex inline'>|\vec{AC}|</code>, and <code class='latex inline'>|\vec{CB}|</code>.</p>
<p>Determine the value of <code class='latex inline'>x</code> and <code class='latex inline'>y</code> in each of the following:</p><p><strong>a)</strong> <code class='latex inline'>3(x, 1) -5(2, 3y) = (11, 33)</code></p><p><strong>b)</strong> <code class='latex inline'>-2(x, x + y) - 3(6, y) = (6, 4)</code></p>
<p> Express the following vectors as a geometric vector by stating its magnitude and direction. </p><p><code class='latex inline'> \displaystyle \vec{u}=(-6\sqrt{3},6) </code></p>
<p>If <code class='latex inline'>\vec{a} = (-60, 11)</code> and <code class='latex inline'>\vec{b}= (-40, -9)</code>, calculate each of the following:</p><p><code class='latex inline'>|\vec{a} + \vec{b}|</code> and <code class='latex inline'>|\vec{a} - \vec{b}|</code></p>
<p>Given <code class='latex inline'>\vec{x} = 2\vec{i}-\vec{j}</code> and <code class='latex inline'>\vec{y}=-\vec{i}+5\vec{j}</code>, determine</p><p><code class='latex inline'>|\vec{x} - \vec{y}|</code></p>
<p> If <code class='latex inline'>\vec{a} = 3\vec{i} + 2\vec{j} - \vec{k}</code> and <code class='latex inline'>\vec{b} = -2\vec{i} + \vec{j}</code>, calculate each magnitude.</p><p><code class='latex inline'> \displaystyle |\vec{a} + \vec{b}| </code></p>
<p>If <code class='latex inline'>\vec{u} = (4, -1)</code> and <code class='latex inline'>\vec{v}= (2, 7)</code>, find</p><p><code class='latex inline'>\displaystyle -4\vec{u} + 7\vec{v} </code></p>
<p>Write the force as a Cartesian vector.</p><p>125 N applied upward</p>
<p>Simplify the following expression.</p><p><code class='latex inline'> \displaystyle (2\hat{i} + 3\hat{j}) + 4(\hat{i} - \hat{j}) </code></p>
<p>Write the force as a Cartesian vector.</p><p>1000 N applied at 72° to the vertical</p>
<p>Find two unit vectors parallel to each vector.</p><p><code class='latex inline'>\displaystyle \vec{a} = [5, -3, 2] </code></p>
<p><strong>a)</strong> For each of the vectors shown below, determine the components of the related position vector.</p><p><strong>b)</strong> Determine the magnitude of each vector.Determine the magnitude of each vector.</p><img src="/qimages/672" />
<p>If <code class='latex inline'>\vec{u} = (4, -1)</code> and <code class='latex inline'>\vec{v}= (2, 7)</code>, find</p><p><code class='latex inline'>\displaystyle 5\vec{v} - 3\vec{v} </code></p>
<p>What is the vector represented in the following diagram?</p><img src="/qimages/2584" />
<p>If <code class='latex inline'>\vec{u}</code> is a vector and <code class='latex inline'>k</code> is a scalar, is it possible that <code class='latex inline'>\vec{u}=k\vec{u}?</code> Under what conditions can this be true?</p>
<p><code class='latex inline'>A(5, 0)</code> and <code class='latex inline'>B(0, 2)</code> are points on the x- and y-axes, respectively. </p> <ul> <li>Find the coordinates of point on the y-axis such that <code class='latex inline'>|\vec{QB}| = |\vec{QA}|</code>.</li> </ul>
<p>Given the vectors <code class='latex inline'>\vec{a}= [-4, 1, 7]</code>, <code class='latex inline'>\vec{b}= [2, 0, -3]</code>, and <code class='latex inline'>\vec{c} =[1, -1, 5]</code>, simplify each vector expression.</p><p><code class='latex inline'>\vec{b} \cdot \vec{c} - \vec{a} \cdot \vec{c}</code> </p>
<p>The initial point of vector <code class='latex inline'>\vec{MN} = [2, 4, -7]</code> is <code class='latex inline'>M(-5, 0, 3)</code>. Determine the coordinates of the terminal point, N.</p>
<p>Resolve <code class='latex inline'>\vec{u} = [3, 4, 7]</code> into two orthogonal vectors, one of which is collinear with <code class='latex inline'>\vec{v} = [1, 2, 3]</code>.</p>
<p><code class='latex inline'>ABCDEFGH</code> is a rectangular prism.</p><p>Write a vector that is equivalent to <code class='latex inline'>\vec{EG} + \vec{GD} + \vec{DE} </code></p>
<p>Write the coordinates of each Cartesian vector. </p><img src="/qimages/1670" />
<p>A cube is constructed form the three vectors <code class='latex inline'>\vec{a}</code>, <code class='latex inline'>\vec{b}</code>, and <code class='latex inline'>\vec{c}</code>, as shown below.</p><img src="/qimages/670" /><p>Is <code class='latex inline'>|\vec{AG}|= |\vec{BH}|</code>? Explain.</p>
<p>Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following: </p><p><code class='latex inline'>|\vec{AB}|</code></p>
<p><code class='latex inline'>ABCDEFGH</code> is a rectangular prism.</p><p>Is it true that <code class='latex inline'>|\vec{HB}| = |\vec{GA}|</code>? Explain.</p>
<p>If <code class='latex inline'>\vec{a} = (-60, 11)</code> and <code class='latex inline'>\vec{b}= (-40, -9)</code>, calculate each of the following:</p><p>a) <code class='latex inline'>|\vec{a}|</code> and <code class='latex inline'>|\vec{b}|</code></p>
<p>Show that the definitions of vector addition and scalar multiplication are consistent by drawing an example to show that <code class='latex inline'>\vec{u}+\vec{u}+\vec{u}+\vec{u}=4\vec{u}</code>.</p>
<p>How would you represent a general vector with its head on the z-axis and its tail at the origin?</p>
<p>Given the point <code class='latex inline'>A(-2, -6, 3)</code> and <code class='latex inline'>B(3, -4, 12)</code>, determine each of the following: </p><p><code class='latex inline'>|\vec{BA}|</code></p>
<p>You are given the vector <code class='latex inline'>\vec{v}=[5,-1]</code>.</p><p>a) State the vertical and horizontal vector components of <code class='latex inline'>\vec{v}</code>.</p><p>b) Find two unit vectors that are collinear with <code class='latex inline'>\vec{v}</code>.</p><p>c) An equivalent vector <code class='latex inline'>\vec{PQ}</code> has its initial point at <code class='latex inline'>P(-2,-7)</code>. Determine the coordinates of <code class='latex inline'>Q</code>.</p><p>d) An equivalent vector <code class='latex inline'>\vec{LM}</code> has its terminal point at <code class='latex inline'>M(5,8)</code>. Determine the coordinates of <code class='latex inline'>L</code>.</p>
<p>Find a single vector equivalent to each of the following:</p><p><code class='latex inline'>2(-2, 3) + (2, 1)</code></p>
<p> Express the following vectors as an algebraic vector in the form <code class='latex inline'>(a,b)</code>. </p><p><code class='latex inline'> \begin{array}{llllll} & \mid \vec{w} \mid=16, \theta=190^{\circ} \end{array} </code></p>
<p>Plot the following points in <code class='latex inline'>\mathbb{R}^3</code>, using a rectangular prism to illustrate each coordinate.</p><p> <code class='latex inline'>F(1, -1, -1)</code></p>
<p>Given the points P(-6, 1), Q(-2, -1) and R-3, 4), find</p><p>a) <code class='latex inline'> \displaystyle \vec{RP} </code></p><p>b) the perimeter of <code class='latex inline'> \displaystyle \triangle PQR </code></p>
<p>Determine the exact magnitude of of <code class='latex inline'>\vec{AB}</code>.</p><p>a) <code class='latex inline'>A(2, 1, 3), B(5, 7, 1)</code></p><p>b) <code class='latex inline'>A(-1, -7, 2), B(-,3 -2, 5)</code></p><p>c) <code class='latex inline'>A(0, 0, 6), B(0, -5, 0)</code></p><p>d) <code class='latex inline'>A(3, -4, 1), B(6, -1, 5)</code></p>
<p>Given <code class='latex inline'>\vec{x} = 2\vec{i}-\vec{j}</code> and <code class='latex inline'>\vec{y}=-\vec{i}+5\vec{j}</code>, determine</p><p><code class='latex inline'>|2\vec{x} - 3\vec{y}|</code></p>
<p>If <code class='latex inline'>\vec{u} = (4, -1)</code> and <code class='latex inline'>\vec{v}= (2, 7)</code>, find</p><p><code class='latex inline'>\displaystyle \vec{u} + \vec{v} </code></p>
<p>Three vectors, <code class='latex inline'>\vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c)</code>, and <code class='latex inline'>\vec{OC} = (0, b, c)</code>, are given. </p><p>Write each of the given vectors using the standard unit vectors.</p>
<p> Given <code class='latex inline'>\vec{p} = 2\vec{i} - \vec{j} + \vec{k}</code> and <code class='latex inline'>\vec{q} = -\vec{i} -\vec{j} + \vec{k}</code>, determine the following in terms of the standard unit vectors. </p><p><code class='latex inline'>\displaystyle \vec{p} + \vec{q}</code></p>
<p>Draw the vector <code class='latex inline'>\vec{AB}</code> joining each pair of points. Then, write the vector in the form [x, y, z]</p><p><code class='latex inline'>A(3, -4, 1), B(6, -1, 5)</code></p>
<p>Simplify each of the following algebraically. </p><p>a) <code class='latex inline'>\vec{u}+\vec{u}+\vec{u}</code></p><p>b) <code class='latex inline'>2\vec{u}-3\vec{v}-3\vec{u}+\vec{v}</code></p><p>c) <code class='latex inline'>3(\vec{u}+\vec{v})-3(\vec{u}-\vec{v})</code></p><p>d) <code class='latex inline'>3\vec{u}+2\vec{v}-2(\vec{v}-\vec{u})+(-3\vec{v})</code></p><p>e) <code class='latex inline'>-(\vec{u}+\vec{v}-4(\vec{u}-2\vec{v})</code></p><p>f) <code class='latex inline'>2(\vec{u}+\vec{v})-2(\vec{u}+\vec{v})</code></p>
<p>Given <code class='latex inline'>\vec{x} = 2\vec{i}-\vec{j}</code> and <code class='latex inline'>\vec{y}=-\vec{i}+5\vec{j}</code>, determine</p><p><code class='latex inline'>|3\vec{y} - 2\vec{x}|</code></p>
<p>For the point following, draw the x-axis, y-axis, and z-axis and accurately draw the position vectors. </p><p><code class='latex inline'> \displaystyle T(-6,1,-8) </code></p>
<p>Write the force as a Cartesian vector.</p><p>500 N applied at 30° to the horizontal</p>
<p><strong>a)</strong> The points <code class='latex inline'>A(5, b, c)</code> and <code class='latex inline'>B(a, -3, 8)</code> are located at the same point in <code class='latex inline'>\mathbb{R}^3</code>. What are the values of <code class='latex inline'>a, b</code>, and <code class='latex inline'>c</code>?</p><p><strong>b)</strong> Write the vector corresponding to <code class='latex inline'>\vec{OA}</code>.</p>
<p>Name the vector <code class='latex inline'>\vec{OA}</code> associated with point <code class='latex inline'>A</code>.</p>
<p>Where is the following general point located? Express your answer with either an axis or plane.</p><p><code class='latex inline'> C(0,y,z) </code></p>
<p>Prove that <code class='latex inline'>k\vec{u} = [ku_1, ku_2, ku_3]</code> for any vector <code class='latex inline'>\vec{u} = [u_1, u_2, u_3]</code> and any scalar <code class='latex inline'>k \in \mathbb{R}</code>.</p>
<p>Draw a set of <code class='latex inline'>x-, y-</code>, and <code class='latex inline'>z-</code>axes and plot the following points:</p><p> <code class='latex inline'>B(0, -2, 0)</code></p>
<p>Draw the vector <code class='latex inline'>\vec{AB}</code> joining each pair of points. Then, write the vector in the form [x, y, z]</p><p><code class='latex inline'>A(2, 1, 3), B(5, 7, 1)</code></p>
How did you do?
Found an error or missing video? We'll update it within the hour! 👉
Save videos to My Cheatsheet for later, for easy studying.