5. Q5c
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Similar Question 1
<p>Consider two vectors <code class='latex inline'>\vec{a}</code> and <code class='latex inline'>\vec{b}</code>.</p><p> In a single diagram, illustrate both <code class='latex inline'>|\vec{a}\times\vec{b}|</code> and <code class='latex inline'>\vec{a}\cdot\vec{b}</code></p>
Similar Question 2
<p>Consider the parallelogram with vertices (0, 0), (3, 0), (5, 3), and (2. 3). Find the angles at which the diagonals of the parallelogram intersect.</p>
Similar Question 3
<p>Consider your answer to question 25. Resolve <code class='latex inline'>\displaystyle \vec{F}=[25,18] </code> into perpendicular components, one of which is in the direction of <code class='latex inline'>\displaystyle \vec{u}=[2,5] </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>The points P(—2, l), Q(—6, 4). and R(4, 3) are three vertices of parallelogram PQRS.</p><p>c) Find the measures of the angles between the diagonals of the parallelogram, to the nearest degree, using your answer from part a)</p><p><code class='latex inline'>\to</code> <strong>a)</strong> Find the coordinates of S.</p>
<p>A force, <code class='latex inline'>\vec{f}</code>, of <code class='latex inline'>25N</code> is acting in the direction of <code class='latex inline'>\vec{a}=[6,1]</code>.</p><p>b) Find the Cartesian vector representing the force, <code class='latex inline'>\vec{f}</code></p>
<p>Calculate the angle between the vectors in each pair. Illustrate geometrically. </p><p><code class='latex inline'>\vec{p}=[7,8]</code>, <code class='latex inline'>\vec{q}=[4,3]</code></p>
<p> A factory worker pushes a package along a broken conveyor belt from <code class='latex inline'>(-4,0)</code> to <code class='latex inline'>(4,0)</code> with a <code class='latex inline'>50</code> <code class='latex inline'>N</code> force at a <code class='latex inline'>30^\circ</code> angle to the conveyor belt. How much mechanical work is done if the units of the conveyor belt are metres?</p>
<p>Determine the work done by the force, <code class='latex inline'>\vec{F}</code>, in the direction of the displacement, <code class='latex inline'>\vec{s}</code>.</p><img src="/qimages/154864" />
<p>Determine the interior angles of <code class='latex inline'>\bigtriangleup</code>ABC for A(5,1), B(4,-7), and C(-1,-8).</p>
<p>The town of Oceanside lies at sea level and the town of Seaview is at an altitude of <code class='latex inline'>\displaystyle 84 \mathrm{~m} </code>, at the end of a straight, smooth road that is <code class='latex inline'>\displaystyle 2.5 \mathrm{~km} </code> long. Following an automobile accident, a tow truck is pulling a car up the road using a force, in newtons, defined by the vector <code class='latex inline'>\displaystyle \vec{F}=\left[\begin{array}{ll}30 & 000,18 & 000\end{array}\right] </code></p><p>a) Find the force drawing the car up the hill and the force, perpendicular to the hill, tending to lift it.</p><p>b) What is the work done by the tow truck in pulling the car up the hill?</p><p>c) What is the work done in raising the altitude of the car?</p><p>d) Explain the differences in your answers to parts b) and c).</p>
<p>Is the following statement true or false? &quot;If <code class='latex inline'>\vec{e}\times\vec{f}=\vec{0}</code>, then <code class='latex inline'>\vec{e}\cdot\vec{f}=\vec{0}</code>.&quot; Justify your response. </p>
<p>Determine the projection of <code class='latex inline'>\vec{u}</code> on <code class='latex inline'>\vec{v}</code>.</p><img src="/qimages/154867" />
<p>A square is defined by the unit vectors <code class='latex inline'>\vec{i}</code> and <code class='latex inline'>\vec{j}</code>. Find the projections of <code class='latex inline'>\vec{i}</code> and <code class='latex inline'>\vec{j}</code> on each of the diagonals of the square</p>
<p>Given <code class='latex inline'>\vec{a}=[-2,3,5]</code>, <code class='latex inline'>\vec{b}=[4,0,-1]</code>, and <code class='latex inline'>\vec{c}=[2,-2,3]</code>, evaluate each expression. </p><p> <code class='latex inline'>\vec{a}\cdot\vec{b}\times\vec{c}</code></p>
<p>Determine the work done by each force, <code class='latex inline'>\vec{F}</code>, in newtons, for an object moving along the vector, <code class='latex inline'>\vec{s}</code>, in metres.</p><p>c) <code class='latex inline'>\vec{F}=[67.8,3.9]</code>, <code class='latex inline'>\vec{s}=[4.7,3.2]</code></p>
<p>The points P(—2, l), Q(—6, 4). and R(4, 3) are three vertices of parallelogram PQRS.</p><p>b) Find the measures of the interior angles of the parallelogram, to the nearest degree, using your answer from part a)</p><p><code class='latex inline'>\to</code> <strong>a)</strong> Find the coordinates of S.</p>
<p>In light reflection, the angle of incidence is equal to the angle of reflection. Let <code class='latex inline'>\displaystyle \vec{u} </code> be the unit vector in the direction of incidence. Let <code class='latex inline'>\displaystyle \vec{v} </code> be the unit vector in the direction of reflection. Let <code class='latex inline'>\displaystyle \vec{w} </code> be the unit vector perpendicular to the face of the reflecting surface. Show that <code class='latex inline'>\displaystyle \vec{v}=\vec{u}-2(\vec{u} \cdot \vec{w}) \vec{w} </code></p><img src="/qimages/154872" />
<p>Draw a diagram to illustrate the meaning of each projection. </p><p>b) <code class='latex inline'>proj_{\vec{u}}(proj_{\vec{v}}\vec{u})</code></p>
<p>A car enters a curve on a highway. If the highway is banked <code class='latex inline'>\displaystyle 10^{\circ} </code> to the horizontal in the curve, show that the vector <code class='latex inline'>\displaystyle \left[\cos 10^{\circ}, \sin 10^{\circ}\right] </code> is parallel to the road surface. The <code class='latex inline'>\displaystyle x </code>-axis is horizontal but perpendicular to the road lanes, and the <code class='latex inline'>\displaystyle y </code>-axis is vertical. If the car has a mass of <code class='latex inline'>\displaystyle 1000 \mathrm{~kg} </code>, find the component of the force of gravity along the road vector. The projection of the force of gravity on the road vector provides a force that helps the car turn. (Hint: The force of gravity is equal to the mass times the acceleration due to gravity.)</p>
<p>Given that <code class='latex inline'>\vec{w}=k\vec{u}+m\vec{v}</code>, where <code class='latex inline'>k,m\in\mathbb{R}</code>, prove algebraically that <code class='latex inline'>\vec{u}\times\vec{v}\cdot\vec{w}=0</code>.</p>
<p>Calculate the angle between the vectors in each pair. Illustrate geometrically. </p><p>d) <code class='latex inline'>\vec{e}=[2,3]</code>, <code class='latex inline'>\vec{f}=[9,-6]</code></p>
<p>Determine the projection, and its magnitude, of <code class='latex inline'>\vec{u}</code> on <code class='latex inline'>\vec{v}</code>.</p><p> <code class='latex inline'>\vec{u}=[3,1,4]</code>, <code class='latex inline'>\vec{v}=[6,2,7]</code></p>
<p>a) Given vectors <code class='latex inline'>\displaystyle \vec{a}=[6,5] </code> and <code class='latex inline'>\displaystyle \vec{b}=[1,3] </code>, find <code class='latex inline'>\displaystyle \operatorname{proj}_{\vec{b}} \vec{a} </code></p><p>b) Resolve <code class='latex inline'>\displaystyle \vec{a} </code> into perpendicular components, one of which is in the direction of <code class='latex inline'>\displaystyle \vec{b} </code></p>
<p>a) Show that the quadrilateral with vertices at P(0,2,5), Q(1,6,2), R(7,4,2), and S(6,0,5) is a parallelogram. </p><p>b) Calculate its area. </p><p>c) Is this parallelogram a rectangle? Explain. </p>
<p>Consider the parallelogram with vertices (0, 0), (3, 0), (5, 3), and (2. 3). Find the angles at which the diagonals of the parallelogram intersect.</p>
<p>A force, <code class='latex inline'>\vec{f}</code>, of <code class='latex inline'>25N</code> is acting in the direction of <code class='latex inline'>\vec{a}=[6,1]</code>.</p><p>Find a unit vector in the direction of <code class='latex inline'>\vec{a}</code></p>
<p>In each case, determine the projection of the first vector on the second. Sketch each projection. </p><p><code class='latex inline'>\vec{g}=[10,-3]</code>, <code class='latex inline'>\vec{h}=[4,-4]</code></p>
<p>Determine the work done by each force, <code class='latex inline'>\vec{F}</code>, in newtons, for an object moving along the vector, <code class='latex inline'>\vec{s}</code>, in metres.</p><p>a) <code class='latex inline'>\vec{F}=[5,2]</code>, <code class='latex inline'>\vec{s}=[7,4]</code></p>
<p>Given <code class='latex inline'>\vec{u}=[2,2,3]</code>, <code class='latex inline'>\vec{v}=[1,3,4]</code>, and <code class='latex inline'>\vec{w}=[6,2,1]</code>, evaluate each expression.</p><p><code class='latex inline'>\vec{u}\times\vec{v}\cdot\vec{v}\times\vec{w}</code></p>
<p>Calculate the angle between the vectors in each pair. Illustrate geometrically. </p><p><code class='latex inline'>\vec{r}=[-2,-8]</code>, <code class='latex inline'>\vec{s}=[6,-1]</code></p>
<p>Show that <code class='latex inline'>|proj_{\vec{u}}\vec{v}|=|\vec{v}|\cos{\theta}</code> can be written as <code class='latex inline'>|proj_{\vec{u}}\vec{v}|=\left(\displaystyle{\frac{\vec{v}\cdot\vec{u}}{\vec{u}\cdot\vec{u}}}\right)\vec{u}</code>.</p>
<p>A triangle has vertices A<code class='latex inline'>(-2,1,3)</code>, B<code class='latex inline'>(7,8,-4)</code>, and <code class='latex inline'>C(5,0,2)</code>. Determine the area of <code class='latex inline'>\triangle</code>ABC.</p>
<p>Determine the projection of <code class='latex inline'>\vec{u}</code> on <code class='latex inline'>\vec{v}</code>.</p><img src="/qimages/154866" />
<p>A store sells digital music players and DVD players. Suppose 42 digital music players are sold at $115 each and 23 DVD players are sold at$95 each. The vector <code class='latex inline'>\vec{a}=[42,23]</code> can be called the sales vector and <code class='latex inline'>\vec{b}=[115,95]</code> the price vector. Find <code class='latex inline'>\vec{a}\cdot\vec{b}</code> and interpret its meaning.</p>
<p>Math Contest The side lengths of a right triangle are in the ratio <code class='latex inline'>\displaystyle 3: 4: 5 </code>. If the length of one of the three altitudes is <code class='latex inline'>\displaystyle 60 \mathrm{~cm} </code>, what is the greatest possible area of this triangle?</p>
<p>Given <code class='latex inline'>\vec{u}=[2,2,3]</code>, <code class='latex inline'>\vec{v}=[1,3,4]</code>, and <code class='latex inline'>\vec{w}=[6,2,1]</code>, evaluate each expression.</p><p><code class='latex inline'>\vec{u}\times\vec{v}\cdot\vec{u}\times\vec{w}</code></p>
<p>Given <code class='latex inline'>\vec{u}=[2,2,3]</code>, <code class='latex inline'>\vec{v}=[1,3,4]</code>, and <code class='latex inline'>\vec{w}=[6,2,1]</code>, evaluate each expression.</p><p><code class='latex inline'>|\vec{u}\times\vec{v}|^2-(\vec{w}\cdot\vec{w})^2</code></p>
<p>The figure shown is a regular decagon with side length 1 cm. Determine the exact value of <code class='latex inline'>x</code></p><img src="/qimages/1684" />
<p>A force, <code class='latex inline'>\vec{F}=[3,5,12]</code>, in newtons, is applied to lift a box, with displacement, <code class='latex inline'>\vec{s}</code>, in metres as given. Calculate the work against gravity and compare it to the work in the direction of travel.</p><p><code class='latex inline'>\vec{s}=[2,1,6]</code></p>
<p>Determine the projection, and its magnitude, of <code class='latex inline'>\vec{u}</code> on <code class='latex inline'>\vec{v}</code>.</p><p><code class='latex inline'>\vec{u} = -2\vec{i} - 7\vec{j} + 3\vec{k}, \vec{v} = 6\vec{i} + \vec{j} -8\vec{k}</code></p>
<p>Find the volume of each parallelepiped, defined by the vectors <code class='latex inline'>\vec{u}</code>, <code class='latex inline'>\vec{v}</code>, and <code class='latex inline'>\vec{w}</code>.</p><p> <code class='latex inline'>\vec{u}=[-2,5,1]</code>, <code class='latex inline'>\vec{v}=[3,-4,2]</code>, and <code class='latex inline'>\vec{w}=[1,3,5]</code></p>
<p>Consider two vectors <code class='latex inline'>\vec{a}</code> and <code class='latex inline'>\vec{b}</code>.</p><p> In a single diagram, illustrate both <code class='latex inline'>|\vec{a}\times\vec{b}|</code> and <code class='latex inline'>\vec{a}\cdot\vec{b}</code></p>
<p>Draw a diagram to illustrate the meaning of each projection. </p><p><code class='latex inline'>proj_{\vec{v}}(proj_{\vec{v}}\vec{u})</code></p>
<p>In each case, determine the projection of the first vector on the second. Sketch each projection. </p><p> <code class='latex inline'>\vec{c}=[2,7]</code>, <code class='latex inline'>\vec{d}=[-4,3]</code></p>
<p>Determine the projection, and its magnitude, of <code class='latex inline'>\vec{u}</code> on <code class='latex inline'>\vec{v}</code>.</p><p> <code class='latex inline'>\vec{u}=[5,-4,8]</code>, <code class='latex inline'>\vec{v}=[3,7,6]</code></p>
<p>Determine the work done by the force, <code class='latex inline'>\vec{F}</code>, in the direction of the displacement, <code class='latex inline'>\vec{s}</code>.</p><img src="/qimages/154863" />
<p>The ramp to the loading dock at a car parts plant is inclined at 20<code class='latex inline'>^\circ</code> to the horizontal. A pallet of parts is moved 5 m up the ramp by a force of 5000 N, at an angle of 15<code class='latex inline'>^\circ</code> to the surface of the ramp.</p><p>a) What information do you need to calculate the work done in moving the pallet along the ramp? </p><p>b) Calculate the work done.</p>
<p>In each case, determine the projection of the first vector on the second. Sketch each projection. </p><p><code class='latex inline'>\vec{e}=[-2,-5]</code>, <code class='latex inline'>\vec{f}=[-5,1]</code></p>
<p>Draw a diagram to illustrate the meaning of each projection. </p><p>a) <code class='latex inline'>proj_{proj_{\vec{v}}\vec{u}}\vec{u}</code></p>
<p>A crate is dragged 3 m along a smooth level floor by a 30 N force, applied at 25<code class='latex inline'>^\circ</code> to the floor. Then, it is pulled 4 m up a ramp inclined at 20<code class='latex inline'>^\circ</code> to the horizontal, using the same force. Then, the crate is dragged a further 5 m along a level platform using the same force again. Determine the total work done in moving the crate.</p>
<p>Determine the work done by each force, <code class='latex inline'>\vec{F}</code>, in newtons, for an object moving along the vector, <code class='latex inline'>\vec{s}</code>, in metres.</p><p>b)<code class='latex inline'>\vec{F}=[100,400]</code>, <code class='latex inline'>\vec{s}=[12,27]</code></p>
<p>Determine the angle between vector <code class='latex inline'>\vec{PQ}</code> and the positive x-axis, given endpoints P(4,7) and Q(8,3).</p>
<p>Show that <code class='latex inline'>|proj_{\vec{u}}\vec{v}|=|\vec{v}|\cos\theta</code> can be written as <code class='latex inline'>\displaystyle{|proj_{\vec{u}}\vec{v}|=\frac{\vec{v}\cdot\vec{u}}{|\vec{u}|}}</code> for <code class='latex inline'>0<\theta<90^\circ</code></p>
<p>Given <code class='latex inline'>\vec{a}=[-2,3,5]</code>, <code class='latex inline'>\vec{b}=[4,0,-1]</code>, and <code class='latex inline'>\vec{c}=[2,-2,3]</code>, evaluate each expression. </p><p><code class='latex inline'>\vec{a}\times\vec{c}\cdot\vec{b}</code></p>
<p>A wrench is rotated with torque of magnitude <code class='latex inline'>100</code> N<code class='latex inline'>\cdot</code>m. The force is applied <code class='latex inline'>30</code> cm from the fulcrum, at an angle of <code class='latex inline'>40^\circ</code>. What is the magnitude of the force, to one decimal place?</p>
<p>Describe when each of the following is true, and illustrate with an example.</p><p>a) <code class='latex inline'>\displaystyle \operatorname{proj}_{\vec{v}} \vec{u}=\operatorname{proj}_{\vec{u}} \vec{v} </code></p><p>b) <code class='latex inline'>\displaystyle \left|\operatorname{proj}_{\vec{v}} \vec{u}\right|=\left|\operatorname{proj}_{\vec{u}} \vec{v}\right| </code></p>
<p>Find the volume of each parallelepiped, defined by the vectors <code class='latex inline'>\vec{u}</code>, <code class='latex inline'>\vec{v}</code>, and <code class='latex inline'>\vec{w}</code>.</p><p> <code class='latex inline'>\vec{u}=[1,4,3]</code>, <code class='latex inline'>\vec{v}=[2,5,6]</code>, and <code class='latex inline'>\vec{w}=[1,2,7]</code></p>
<p>Given <code class='latex inline'>\vec{a}=[-2,3,5]</code>, <code class='latex inline'>\vec{b}=[4,0,-1]</code>, and <code class='latex inline'>\vec{c}=[2,-2,3]</code>, evaluate each expression. </p><p> <code class='latex inline'>\vec{b}\cdot\vec{a}\times\vec{c}</code></p>
<p>Determine the work done by the force, <code class='latex inline'>\vec{F}</code>, in the direction of the displacement, <code class='latex inline'>\vec{s}</code>.</p><img src="/qimages/154865" />
<p>Determine the work done by the force, <code class='latex inline'>\vec{F}</code>, in the direction of the displacement, <code class='latex inline'>\vec{s}</code>.</p><p><code class='latex inline'>\displaystyle |\vec{F}|=50 \mathrm{~N} </code></p><img src="/qimages/154861" />
<p>A bicycle pedal is pushed by a <code class='latex inline'>75</code>-N force, exerted as shown in the diagram. The shaft of the pedal is <code class='latex inline'>15</code> cm long. Find the magnitude of the torque vector, in newton-metres, about point A.</p><img src="/qimages/1682" />
<p>Given <code class='latex inline'>\vec{u}=[2,2,3]</code>, <code class='latex inline'>\vec{v}=[1,3,4]</code>, and <code class='latex inline'>\vec{w}=[6,2,1]</code>, evaluate each expression.</p><p> <code class='latex inline'>|\vec{u}\times\vec{u}|+\vec{u}\cdot\vec{u}</code></p>
<p>The points P(—2, l), Q(—6, 4). and R(4, 3) are three vertices of parallelogram PQRS.</p><p>a) Find the coordinates of S.</p>
<p>A force, <code class='latex inline'>\vec{F}=[3,5,12]</code>, in newtons, is applied to lift a box, with displacement, <code class='latex inline'>\vec{s}</code>, in metres as given. Calculate the work against gravity and compare it to the work in the direction of travel.</p><p><code class='latex inline'>\vec{s}=[0,0,8]</code></p>
<p>Justin applies a force at <code class='latex inline'>20^\circ</code> to the horizontal to move a football tackling dummy $8 m$ horizontally. He does <code class='latex inline'>150</code> <code class='latex inline'>J</code> of mechanical work. What is the magnitude of the force?</p>
<p>Consider your answer to question 25. Resolve <code class='latex inline'>\displaystyle \vec{F}=[25,18] </code> into perpendicular components, one of which is in the direction of <code class='latex inline'>\displaystyle \vec{u}=[2,5] </code>.</p>
<p>A force, <code class='latex inline'>\vec{f}</code>, of <code class='latex inline'>25N</code> is acting in the direction of <code class='latex inline'>\vec{a}=[6,1]</code>.</p><p>c) The force <code class='latex inline'>\vec{f}</code> is exerted on an object moving from point (4, 0) to point (15, 0), with distance in metres. Determine the mechanical work done.</p>
<p>Consider two vectors <code class='latex inline'>\vec{a}</code> and <code class='latex inline'>\vec{b}</code>.</p><p>Interpret <code class='latex inline'>|\vec{a}\times\vec{b}|^2+|\vec{a}\cdot\vec{b}|^2</code> and illustrate it on your diagram. </p>
<p>Three edges of a right triangular prism are defined by the vectors <code class='latex inline'>\vec{a}=[1,3,2]</code>, <code class='latex inline'>\vec{b}=[2,2,4]</code>, and <code class='latex inline'>\vec{c}=[12,0,-6]</code>.</p><p>a) Draw a diagram of the prism, identifying which edge of the prism is defined by each vector.</p><p>b) Determine the volume of the prism.</p><p>c) Explain how your method of solving this problem would change if the prism were not necessarily a right prism.</p>
<p>Determine the projection, and its magnitude, of <code class='latex inline'>\vec{u}</code> on <code class='latex inline'>\vec{v}</code>.</p><p><code class='latex inline'>\vec{u} = \vec{i} - \vec{k}</code>, <code class='latex inline'>\vec{v} = 9\vec{i} + \vec{j}</code></p>
<p>A <code class='latex inline'>65</code>-kg boy is sitting on a see saw <code class='latex inline'>0.6</code> m from the balance point.How far from the balance point should a <code class='latex inline'>40</code>-kg girl sit so that the see saw remains balanced?</p>
<p>A force, <code class='latex inline'>\vec{F}=[3,5,12]</code>, in newtons, is applied to lift a box, with displacement, <code class='latex inline'>\vec{s}</code>, in metres as given. Calculate the work against gravity and compare it to the work in the direction of travel.</p><p><code class='latex inline'>\vec{s}=[2,0,10]</code></p>
<p>In each case, determine the projection of the first vector on the second. Sketch each projection. </p><p><code class='latex inline'>\vec{a}=[6,-1]</code>, <code class='latex inline'>b=[11,5]</code></p>
<p>When a wrench is rotated. the magnitude of the torque is <code class='latex inline'>10</code> N<code class='latex inline'>\cdot</code>m. An <code class='latex inline'>80</code>-N force is applied <code class='latex inline'>20</code> cm from the fulcrum. At what angle to the wrench is the force applied?</p>
<p>Find the volume of each parallelepiped, defined by the vectors <code class='latex inline'>\vec{u}</code>, <code class='latex inline'>\vec{v}</code>, and <code class='latex inline'>\vec{w}</code>.</p><p><code class='latex inline'>\vec{u}=[1,1,9]</code>, <code class='latex inline'>\vec{v}=[0,0,4]</code>, and <code class='latex inline'>\vec{w}=[-2,0,5]</code></p>
<p>How much work is done against gravity by a worker who carries a 25 kg carton up a 6 m long set of stairs, inclined at 30<code class='latex inline'>^\circ</code>?</p>
<p>An axle has two wheels of radii <code class='latex inline'>0.75</code> m and <code class='latex inline'>0.35</code> m attached to it. A <code class='latex inline'>10</code>-N force is applied horizontally to the edge of the larger wheel and a <code class='latex inline'>5</code>-N weight hangs from the edge of the smaller wheel. What is the net torque acting on the axle?</p><img src="/qimages/1683" />
<p>A superhero pulls herself 15 m up the side of a wall with a force of 500 N, at an angle of 12<code class='latex inline'>^\circ</code> to the vertical. What is the work done?</p>
<p>Consider two vectors <code class='latex inline'>\vec{a}</code> and <code class='latex inline'>\vec{b}</code>.</p><p>Show that <code class='latex inline'>|\vec{a}\times\vec{b}|^2+|\vec{a}\cdot\vec{b}|^2=|\vec{a}|^2|\vec{b}|^2</code>. </p>
<p>Calculate the angle between the vectors in each pair. Illustrate geometrically. </p><p><code class='latex inline'>\vec{t}=[-7,2]</code>, <code class='latex inline'>\vec{u}=[6,11]</code></p>
<p>Given <code class='latex inline'>\vec{a}=[-2,3,5]</code>, <code class='latex inline'>\vec{b}=[4,0,-1]</code>, and <code class='latex inline'>\vec{c}=[2,-2,3]</code>, evaluate each expression. </p><p><code class='latex inline'>\vec{a}\times\vec{b}\cdot\vec{c}</code></p>
<p>A force of <code class='latex inline'>90</code> N is applied to a wrench in a counterclockwise direction at <code class='latex inline'>70^\circ</code> to the handle, <code class='latex inline'>15</code> cm from the centre of the bolt.</p><p>a) Calculate the magnitude of the torque.</p><p>b) In what direction does the bolt move?</p><img src="/qimages/8005" />
<p>Let <code class='latex inline'>\vec{u}</code>, <code class='latex inline'>\vec{v}</code> and <code class='latex inline'>\vec{w}</code> be mutually orthogonal vectors. What can be said about <code class='latex inline'>\vec{u}+\vec{v}</code>, <code class='latex inline'>\vec{u}+\vec{w}</code>, and <code class='latex inline'>\vec{v}+\vec{w}</code></p>
<p>Determine the projection of <code class='latex inline'>\vec{u}</code> on <code class='latex inline'>\vec{v}</code>.</p><img src="/qimages/154868" />
<p>Prove that <code class='latex inline'>(\vec{a}\times\vec{b})\times\vec{c}</code> is in the same plane as <code class='latex inline'>\vec{a}</code> and <code class='latex inline'>\vec{b}</code>.</p>
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