Introduction to Vectors Chapter Review
Chapter
Chapter 6
Section
Introduction to Vectors Chapter Review
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Solutions 38 Videos

Determine whether each of the following statements is true or false. Provide a brief explanation for each answer.

a. |\vec{a} + \vec{b}| \geq |\vec{a}|

b. |\vec{a} + \vec{b}| = |\vec{a} + \vec{c}| implies |\vec{b}| = |\vec{c}|

c. \vec{a} + \vec{b} = \vec{a} + \vec{c} implies \vec{b} = \vec{c}

d. \vec{RF} = \vec{SW} implies \vec{RS}= \vec{FW}

e. m\vec{a} + n\vec{a} = (m + n)\vec{a}

f. If |\vec{a}| = |\vec{b}| and |\vec{c}| = |\vec{d}|, then |\vec{a} + \vec{b}| =|\vec{c} + \vec{d}|.

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Q1

If \vec{x} = 2\vec{a} - 3\vec{b} -4\vec{c}, \vec{y} = -2\vec{a} + 3\vec{b} + 3\vec{c}, and \vec{z} = 2\vec{a} - 3\vec{b} + 5\vec{c}, determine simplified expressions for the following:

\displaystyle 2\vec{x} - 3\vec{y} + 5\vec{z}

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1.54mins
Q2a

If \vec{x} = 2\vec{a} - 3\vec{b} -4\vec{c}, \vec{y} = -2\vec{a} + 3\vec{b} + 3\vec{c}, and \vec{z} = 2\vec{a} - 3\vec{b} + 5\vec{c}, determine simplified expressions for the following:

\displaystyle 3(-2\vec{x} -4\vec{y} + \vec{z}) -(2\vec{x} -\vec{y} + \vec{z}) -2(-4\vec{x} -5\vec{y} + \vec{z})

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Q2b

If X(-2, 1, 2) and Y(-4, 4, 8) are two points in \mathbb{R^3}, determine the following:

a. \displaystyle \vec{XY} and \displaystyle |\vec{XY} |

b. The coordinates of a unit vector in the same direction as \vec{XY}.

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3.20mins
Q3

X(-1, 2, 6) and Y(5, 5, 12) are two points in \mathbb{R^3}

a. Determine the components of a position vector equivalent to \vec{YX}.

b. Determine the components of a unit vector that is in the same direction as \vec{YX}.

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2.41mins
Q4

Find the components of the unit vector with the opposite direction to that of the vector from M(2, 3, 5) to N(8, 1, 2).

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2.15mins
Q5

A parallelogram has its sides determined by the vectors \vec{OA} = (3, 2, -6) and \vec{OB} = (-6, 6, -2).

a. Determine the components of the vectors representing the diagonals.

b. Determine the angles between the sides of the parallelogram.

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4.52mins
Q6

The points A(-1, 1, 1), B(2, 0, 3), and C(3, 3, -4) are vertices of a triangle.

Show that the area of triangle ABC.

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2.27mins
Q7a

The points A(-1, 1, 1), B(2, 0, 3), and C(3, 3, -4) are vertices of a triangle.

Calculate the area of triangle ABC.

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1.23mins
Q7b

The points A(-1, 1, 1), B(2, 0, 3), and C(3, 3, -4) are vertices of a triangle.

Calculate the perimeter of triangle ABC.

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1.09mins
Q7c

The points A(-1, 1, 1), B(2, 0, 3), and C(3, 3, -4) are vertices of a triangle.

Calculate the coordinates of the fourth vertex D that completes the rectangle of which A, B, and C are the other three vertices.

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3.31mins
Q7d

The vectors \vec{a}, \vec{b}, and \vec{c} are as shown.

a. Construct the vector \vec{a} -\vec{b} + \vec{c}.

b. If the vectors \vec{a} and \vec{b} are perpendicular, and if |\vec{a}| = 4 and |\vec{b}| = 3, determine |\vec{a}+ \vec{b}|.

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1.48mins
Q8

Given \vec{p} = (-11, 7), \vec{q} = (-3, 1), and \vec{r} = (-1, 2), express each vector as a linear combination of the other two.

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2.26mins
Q9

a. Find an equation to describe the set of points equidistant from A(2, 1, 3) and B(1, 2, -3).

b. Find the coordinates of two points that are equidistant from A and B.

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4.45mins
Q10

Calculate the values of a, b, and c in each of the following:

\displaystyle 2(a, b, 4) + \frac{1}{2}(6, 8, c) -3(7, c, -4) = (-24, 3, 25)

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2.23mins
Q11a

Calculate the values of a, b, and c in each of the following:

\displaystyle 2(a, a, \frac{1}{2}a) +(3b, 0, -5c) + 2(c, \frac{3}{2}c, 0) = (3, -22, 54)

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3.29mins
Q11b

Determine whether the points A(1, -1, 1), B(2, 2, 2), and C(4, -2, 1) represent the vertices of a right triangle.

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1.43mins
Q12a

Determine whether the points P(1, 2, 3), Q(2, 4, 6), and R(-1, -2, -3) are collinear.

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1.25mins
Q12b

a. Show that the points A(3, 0, 4), B(1, 2, 5), and C(2, 1, 3) represent the vertices of a right triangle.

b. Determine \cos \angle ABC.

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4.17mins
Q13

In the following rectangle, vectors are indicated by the direction of the arrows.

a. Name two pairs of vectors that are opposites.

b. Name two pairs of identical vectors.

c. Explain why |\vec{AD}|^2 + |\vec{DC}|^2 = |\vec{DB}|^2

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2.52mins
Q14

A rectangular prism measuring 3 by 4 by 5 is drawn on a coordinate axis as shown in the diagram.

a. Determine the coordinates of points C, P, E, and F.

b. Determine position vectors for \vec{DB} and \vec{CF}.

c. By drawing the rectangle containing \vec{DB} and \vec{OP}, determine the acute angle between these vectors.

d. Determine the angle between \vec{OP} and \vec{AE}.

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5.25mins
Q15

The vectors \vec{a} and \vec{e} are such that |\vec{d}| =3 and |\vec{e}| =5, and the angle between them is 30^o. Determine each of the following:

|\vec{d} + \vec{e}|

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1.33mins
Q16a

The vectors \vec{a} and \vec{e} are such that |\vec{d}| =3 and |\vec{e}| =5, and the angle between them is 30^o. Determine each of the following:

|\vec{d} - \vec{e}|

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1.17mins
Q16b

The vectors \vec{a} and \vec{e} are such that |\vec{d}| =3 and |\vec{e}| =5, and the angle between them is 30^o. Determine each of the following:

|\vec{e} - \vec{d}|

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1.14mins
Q16c

An airplane is headed south at speed 400 km/h. The airplane encounters a wind from the east blowing at 100 km/h.

a. How far will the airplane travel in 3 h?

b. What is the direction of the airplane?

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1.55mins
Q17

Explain why the set of vectors: \{(2, 3), (3, 5)\} span \mathbb{R^2}.

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1.10mins
Q18a

Find m and n in the following: m(2, 3) + n(3, 5) = (323, 795).

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2.05mins
Q18b

Show that the vector \vec{a} = (5, 9, 4) can be written as a linear combination of the vectors \vec{b} and \vec{c}, where \vec{b} = (-2, 3, 1) and \vec{c} = (3, 1, 4). Explain why \vec{a} lies in the plane determined by \vec{b} and \vec{c}.

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2.27mins
Q19a

Is the vector \vec{a}= (-13, 36, 23) in the span of \vec{b} = (-2, 3, 1) and \vec{c} = (3, 1, 4)? Explain your answer.

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2.51mins
Q19b

A cube is placed so that it has three of its edges located along the positive x-, y-, and z-axes (one edge along each axis) and one of its vertices at the origin.

a. If the cube has a side length of 4, draw a sketch of this cube and write the coordinates of its vertices on your sketch.

b. Write the coordinates of the vector with its head at the origin and its tail at the opposite vertex.

c. Write the coordinates of a vector that starts at (4, 4, 4) and is a diagonal in the plane parallel to the xz-plane.

d. What vector starts at the origin and is a diagonal in the xy-plane?

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2.16mins
Q20

If \vec{a} = \vec{i} + \vec{j} - \vec{k}, \vec{b} = 2\vec{i} - \vec{j} + 3\vec{k}, and \vec{c} = 2\vec{i} + 13\vec{k}, determine |2(\vec{a}+\vec{b}-\vec{c}) - (\vec{a} + 2\vec{b} ) + 3(\vec{a} - \vec{b} + \vec{c})|.

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2.36mins
Q21

The three points A( -3, 4), B(3, -4) and C(5, 0) are on a circle with radius 5 and centre at the origin. Points A and B are the endpoints of a diameter, and point C is on the circle.

a. Calculate |\vec{AB}|, |\vec{AC}|, and |\vec{BC}|.

b. Show that A, B, and C are the vertices of a right triangle.

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2.41mins
Q22

In terms of \vec{a}, \vec{b}, \vec{c}, and \vec{0}, find a vector expression for each of the following:

a. \vec{FL}

b. \vec{MK}

c. \vec{HJ}

d. \vec{IH} + \vec{KJ}

e. \vec{IK} - \vec{IH}

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2.00mins
Q23

Draw a diagram showing the vectors \vec{a} and \vec{b}, where |\vec{a}| = 2|\vec{b}| and |\vec{b}| = 2|\vec{b}| and |\vec{b}| = |\vec{a} + \vec{b}| are both true.

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2.08mins
Q24

If the vectors \vec{a} and \vec{b} are perpendicular to each other, express each of the following in terms of |\vec{a}| and |\vec{b}|:

|\vec{a}+ \vec{b}|

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0.53mins
Q25a

If the vectors \vec{a} and \vec{b} are perpendicular to each other, express each of the following in terms of |\vec{a}| and |\vec{b}|:

|\vec{a}- \vec{b}|

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0.48mins
Q25b

If the vectors \vec{a} and \vec{b} are perpendicular to each other, express each of the following in terms of |\vec{a}| and |\vec{b}|:

|2\vec{a}+ 3\vec{b}|

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1.14mins
Q25c

Show that if \vec{a} is perpendicular to each of the vectors \vec{b} and \vec{c}, then \vec{a} is perpendicular to 2\vec{b} + 4\vec{c}.

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1.21mins
Q26