Free Printable Sections 13-1 to 13-4 Review

1.

Which are equivalent vectors?

a)

u and -u

b)

u and a

c)

u and v

d)

u and b

2.

Given an initial point R ( -12, 4 ) and terminal point S ( 28, -17 ), find the component form of the vector RS.

a)

〈14,-13〉

b)

〈-14,13〉

c)

〈-40,-21〉

d)

〈40,-21〉

3.

Given an initial point A ( 2, 4 ) and terminal point B ( -8, 7 ), find the component form of the vector AB.

a)

〈-10,3〉

b)

〈-6,3〉

c)

〈10,11〉

d)

〈-6,-3〉

4.

Write in component form.

a)

<0, 4>

b)

<8, 2>

c)

<2, 8>

d)

<0, 2>

5.

Given u = <3, 7> and v =<-5, 4>, find 2v - 3u

a)

<21, 2>

b)

<-19, -13>

c)

<-15, 3>

d)

23

6.

Find the magnitude of the vector <10, -8>

a)

$3\sqrt[]{2}$

b)

$2\sqrt[]{41}$

c)

6

d)

18

7.

Find the magnitude of the vector <2, -3>.

a)

√11

b)

13

c)

√-4

d)

√13

8.

If vectors X and Y are added, then the resultant would be best represented by vector _____.

a)

A

b)

B

c)

C

d)

D

9.

Find the unit vector in the same direction as v.

a)

<-5/13, 12/13>

b)

<5/13, -12/13>

c)

<5/13, 12/13>

d)

<-5/13, -12/13>

10.

Vectors a, b, c are defined as:

a = 2i + 5j; b = i - 3j; c = -3i - j. Answer all correct answers

a)

a + b = 3i + 8j

b)

a - b = i + 8j

c)

a + c = -i + 4j

d)

c - b = -5i + 6j

11.

Given points A(3, 2, -4) and B(2, 1, -1), the vector AB is

a)

i + 3j -5k

b)

-i -3j +3k

c)

i -j -5k

d)

-i -j +3k

12.

Given the vector $v=3i-2j+6k$ , the magnitude of a is:

a)

36

b)

49

c)

7

d)

6

13.

Which Graph shows point (0, 3, -2)

a)

b)

c)

d)

14.

Which Graph shows point (0, -4, -2)

a)

b)

c)

d)

15.

Which Graph shows point (-1, -1, 3)

a)

b)

c)

d)

16.

Which Graph shows point (3, -5, 2)

a)

b)

c)

d)

17.

Which Graph shows point (3, 3, -3)

a)

b)

c)

d)

18.

Write the coordinates of the point:

a)

(0, 1, 1)

b)

(1, 0, -1)

c)

(1, -1, 0)

d)

(-1, -1, 0)

19.

Write the coordinates of the point:

a)

(3, 4, -2)

b)

(3, -4, 2)

c)

(-3, 4, 2)

d)

(-3, -4, 0)

20.

Write the coordinates of the point:

a)

(1, -2, -2)

b)

(1, -2, 2)

c)

(-1, 4, 2)

d)

(-1, 2, 2)

21.

Write the coordinates of the point:

a)

(-2, -2, 1)

b)

(-2, -4, 1)

c)

(-1, 4, -2)

d)

(-1, 2, -4)

22.

which shows the correct point if x<0, y<0 and z<0

a)

b)

c)

d)

23.

which shows the correct point if x<0, y>0 and z<0

a)

b)

c)

d)

24.

Which shows the correct vector if x>0, y>0 and z<0

a)

b)

c)

d)

25.

What is the magnitude of a vector geometrically?

a)

the length of the vector drawn

b)

the arrow head of the vector drawn

c)

The angle from the horizontal axis

d)

The angle from north

26.

Vector is a quantity with:

a)

magnitude and direction

b)

magnitude

c)

direction

d)

none of the above

27.

Find the magnitude of $\overrightarrow{v}$ with component form $<4,5,2>$ .

a)

$3\sqrt{5}$

b)

$\sqrt{11}$

c)

$45$

d)

$11$

28.

Calculate $\overrightarrow{z}=3\overrightarrow{w}-2\overrightarrow{v}+\overrightarrow{u}$ given $\overrightarrow{u}=<-1,3,2>$ , $\overrightarrow{v}=<1,-2,-2>$ , and $\overrightarrow{w}=<5,0,-5>$ .

a)

$<17,-4,11>$

b)

$<16,-1,13>$

c)

$<1,13,5>$

d)

$<12,7,-9>$

29.

Find the unit vector in the direction of $<3,4,-10>$

a)

$<\frac{3\sqrt{\ 5}}{25},\frac{4\sqrt{\ 5}}{25},\frac{-2\sqrt{\ 5}}{5}>$

b)

$<\frac{3}{\sqrt{\ 3}},\frac{4}{\sqrt{\ 3}},\frac{-10}{\sqrt{\ 3}}>$

c)

$<\frac{3\sqrt{\ 5}}{5},\frac{4\sqrt{\ 5}}{5},\frac{-2\sqrt{\ 5}}{5}>$

d)

$<\frac{15\sqrt{\ 5}}{5},\frac{20\sqrt{\ 5}}{5},\frac{-50\sqrt{\ 5}}{5}>$

30.

Write the component form of

$\overrightarrow{v}$ with initial point $\left(-5,3,1\right)$ and terminal point $\left(-7,-1,6\right)$ .

a)

$<-2,-4,5>$

b)

$<2,4,-5>$

c)

$<-2,4,5>$

d)

$<2,-4,5>$

31.

If A = <3, 6> and B = <11, 1>, then the vector A + B in terms of i and j is equal to

a)

3i + 6j

b)

8i - 5j

c)

8i + 5j

d)

14i + 7j

32.

Vector a = 2i + 3j - 5k. What is the value of 3a?

a)

3i + 3j - 3k

b)

5i + 6j - 8k

c)

6i + 9j -15k

d)

6i + 6j - 15k

33.

What is the distance between (-2, 4, 6) and (3, -5, -7)?

a)

$\sqrt{275}$

b)

$\sqrt{251}$

c)

$\sqrt{246}$

d)

$\sqrt{287}$

34.

What is the midpoint between (-2, 4, 6) and (3, -5, -7)?

a)

$\left(\frac{1}{2},\ -\frac{1}{2},\ -\frac{1}{2}\right)$

b)

$\left(\frac{5}{2},\ -\frac{9}{2},\ \frac{13}{2}\right)$

c)

$\left(\frac{1}{2},\ -\frac{9}{2},\ -\frac{13}{2}\right)$

d)

$\left(-\frac{5}{2},-\frac{1}{2},-\frac{1}{2}\right)$

35.

What is the center and radius of $x^2+y^2+z^2-2x-4y+8z-15=0$

a)

(1, 2, -4), r = 6

b)

(-1, -2, 4), r = 6

c)

(-2, -4, -8), r = 36

d)

(1, 2, -4), $r=\sqrt{21}$

36.

What is the component form of the vector from (3, 2, -8) to (-5, 3, 9)?

a)

<-8, 1, 17>

b)

<8, -1, -17>

c)

<-15, 6, -72>

d)

<15, -6, 72>

37.

What is the magnitude of vector v?

a)

$\sqrt{113}$

b)

$\sqrt{209}$

c)

$\sqrt{56}$

d)

$\sqrt{159}$

38.

Which vector has these components?

(3, 2)

a)

b)

c)

d)

39.

What are the vector components?

a)

(2, 2)

b)

(3, 3)

c)

(3, 2)

d)

(2, 3)

40.

What are the vector components?

a)

(-2, -2)

b)

(-3, -3)

c)

(-3, -2)

d)

(-2, -3)

41.

Find the resultant vector.

a)

<5, 3>

b)

<1, 7>

c)

<7, 1>

d)

<3, 5>

42.

Find the components of a vector v such that
v + 〈-6,4〉 = 〈10,-3〉

a)

〈4,1〉

b)

〈16,4〉

c)

〈16,-7〉

d)

none of these

43.

Which of the following would be the correct Equation for a Sphere with a Center at (-1, 0, 3) and a Radius of 10?

a)

(x + 1)² + y² + (z - 3)² = 100

b)

(x - 1)² + y² + (z + 3)² = 100

c)

(z + 1)² - (x + 3)² - y² = 100

44.

What would be the equation for a Sphere with a center of (8,3,7) and a Radius of 14?

a)

(x - 3)² + (y - 8)² + (z - 7)² = 196

b)

(x - 8)² + (y - 3)² + (z - 7)² = 196

c)

(x - 4)² + (y - 1.5)² + (z - 3.5)² = 196

45.

Which of the following is the equation of the sphere with center (8,15,10) and passing through (−14,13,−14)?

a)

(x-8)²+ (y-15)²+ (z-10)²= 56

b)

(x-8)²+ (y-15)²+ (z-10)²= 1064

c)

(x-8)²- (y-15)²+ (z-10)²= 56

d)

(x-8)²- (y-15)²+ (z-10)²= 1064

46.

$\overrightarrow{TU}+\overrightarrow{US}=$

a)

$\overrightarrow{TS}$

b)

$\overrightarrow{ST}$

c)

$\overrightarrow{VS}$

d)

$\overrightarrow{SV}$

47.

Which vector equation matches the vector operation shown in the diagram below?

a)

u-v=w

b)

u-v=-w

c)

u+v=w

d)

v+u=-w

48.

Find the unit vector in the direction of v = <-3,3>.

a)

<-3,3>

b)

$<\frac{-1}{3\sqrt[]{2}},\frac{1}{3\sqrt[]{2}}>$

c)

<-3/18, 3/18>

d)

$<\frac{-1}{\sqrt[]{2}},\frac{1}{\sqrt[]{2}}>$

49.

Find the unit vector in the direction of v = <5,12>.

a)

<5,12>

b)

$<\frac{5}{13},\frac{12}{13}>$

c)

$<\frac{5}{\sqrt[]{13}},\frac{12}{\sqrt[]{13}}>$

d)

$<\frac{5}{\sqrt[]{2}},\frac{12}{\sqrt[]{2}}>$

50.

Write $v=<v_1,\ v_2>$ as a linear combination of i and j.

a)

$v_1\ +\ v_2$

b)

$v_1i\ +\ v_2i$

c)

$v_1i\ +\ v_2j$

d)

$v_1j\ +\ v_2i$

51.

Let v be the vector with initial point (2,-5) and terminal point (-1,3). Write v as a linear combination of the standard unit vectors..

a)

3i + 8j

b)

-3i - 8j

c)

-3i + 8j

d)

3i-8j

52.

A unit vector is a vector that is _____________the given vector and has _____________.

a)

parallel to, magnitude of 1

b)

equal to, the same direction

c)

opposite, the same size

d)

smaller than, the same direction

53.

$u\cdot v$

a)

-82

b)

-58

c)

82

d)

58

54.

$2w\cdot3v$

a)

414

b)

594

c)

-414

d)

504

55.

Find the angle between v and w.

a)

$21.8\degree$

b)

$111.8\degree$

c)

$68.2\degree$

d)

$158.2\degree$

56.

A heavy crate is dragged 50 feet along a level floor. Find the work done if a force of 30 pounds at an angle of 42 degrees with the horizontal is used.

a)

1818.65

b)

809.912

c)

1114.72

d)

1003.70

57.

Find the dot product of $<1,-2>$ and $<3,2>$ .

a)

1

b)

$<3,-4>$

c)

$<4,0>$

d)

$-1$

58.

Find the angle between vectors $<1,3>$ and $<2,-5>$ .

a)

$40.24°$

b)

$49.76°$

c)

$139.76°$

d)

$92.57°$

59.

If the dot product of two nonzero vectors is equal to zero, then what do we know about the two vectors?

a)

They are parallel.

b)

They are orthogonal.

c)

They are neither parallel nor orthogonal.

60.

What type of quantity is produced by the dot product of two vectors?

a)

scalar

b)

vector

61.

Find $v\cdot v$ given $v=4i-3j$

a)

5

b)

7

c)

25

d)

16

62.

Are u and v orthogonal?

$u\ =\ <2,\ -3>$

v\ =\ <-6,\ -4>

a)

Yes

b)

No

63.

Which of the following vector(s) is/are orthogonal to $<-2,\ 3>$

a)

$<3,\ -2>$

b)

$<-3,\ -2>$

c)

$<1.5,\ 1>$

d)

$<\frac{3}{4},\ \frac{1}{2}>$

64.

Are the vectors orthogonal, parallel or neither: $u=<-12,30>$ and $v=<\frac{1}{2},-\frac{5}{4}>$

a)

Parallel

b)

Orthogonal

c)

Neither

65.

For what value of x will the two vectors be orthogonal?

$<x,-3>$ and $<4,8>$

a)

$6$

b)

$-6$

c)

$\frac{25}{4}$

d)

$-\frac{25}{4}$

66.

Determine if the vectors are parallel, orthogonal, or neither. $<4,\ -2>$ and $<6,\ 12>$

a)

Parallel

b)

Orthogonal

c)

Neither

67.

Determine if the vectors are parallel, orthogonal, or neither.
$<4,\ -6>$ and $<-8,\ 12>$

a)

Parallel

b)

Orthogonal

c)

Neither

68.

Given $\overrightarrow{u}=<-3,5>,\ \overrightarrow{v}=<2,-1>$ . Calculate $\overrightarrow{u}\cdot\overrightarrow{v}$

a)

10

b)

-1

c)

-11

d)

9

69.

In the coordinate plane, given points A(1,4), B(-2,3), C(3,0). Find $\overrightarrow{AB}\cdot\overrightarrow{AC}$

a)

-6

b)

2

c)

-2

d)

6

70.

Find the dot product between vectors $\overrightarrow{m}=<-4,0,4>$ and $\overrightarrow{n}=<2,3,-5>$ .

a)

-28

b)

-25

c)

0

d)

12

71.

If $\left|v\right|=8$ and $\left|w\right|=6$ and the angle between them is θ = 120˚ $Find\ \overrightarrow{v}\cdot\overrightarrow{w}$

a)

-24

b)

24

c)

48

d)

-48

72.

What is the angle between two orthogonal vectors?

a)

$180\degree$

b)

$90\degree$

c)

$60\degree$

d)

$45\degree$

73.

If $\overrightarrow{a}=3i\ +5j+k\ \ and\ \overrightarrow{b}=2i+j+3k\ then\ \overrightarrow{a}\cdot\overrightarrow{b}=$

a)

41

b)

12

c)

21

d)

14

74.

Scalar projection of $\overrightarrow{\ \ a\ }\ =2i-3j-5k\ on\ \overrightarrow{\ \ b}=i-j-k\ is$

a)

10

b)

$10\sqrt{3}$

c)

$\frac{10\sqrt{3}}{3}$

d)

--10

75.

If $\overrightarrow{\ \ a\ }=5i-7j-2k\ is\ perpendicular\ to\ \overrightarrow{\ \ b}\ =2i\ +pj+3k\ then\ p\ =$

a)

6/4

b)

7/4

c)

4/6

d)

4/7

76.

Given $\overrightarrow{u}=<1,-2>,\ \overrightarrow{v}=<-4,-3>$ , find $scal_{\overrightarrow{v}}\overrightarrow{u}$

a)

$\frac{2}{5}$

b)

$2$

c)

$\frac{2}{\sqrt[]{5}}$

d)

$-2$

77.

Given $\overrightarrow{u}=<3,0,4>,\ \overrightarrow{v}=<2,3,3>$ , find $scal_{\overrightarrow{v}}\overrightarrow{u}$

a)

$\frac{18}{5}$

b)

$18$

c)

$\frac{18}{\sqrt[]{22}}$

d)

$2$

78.

Given $\overrightarrow{u}=<3,0,4>,\ \overrightarrow{v}=<2,3,3>$ , find $scal_{\overrightarrow{u}}\overrightarrow{v}$

a)

$\frac{18}{5}$

b)

$18$

c)

$\frac{18}{\sqrt[]{22}}$

d)

$2$

79.

Given $\overrightarrow{u}=<3,0,4>,\ \overrightarrow{v}=<2,3,3>$ , find $\operatorname{proj}_{\overrightarrow{v}}\overrightarrow{u}$

a)

$\frac{18}{25}<3,0,4>$

b)

$\frac{18}{22}<3,0,4>$

c)

$\frac{18}{22}<2,3,3>$

d)

$\frac{18}{25}<2,3,3>$

80.

Given $\overrightarrow{u}=<3,0,4>,\ \overrightarrow{v}=<2,3,3>$ , find $\operatorname{proj}_{\overrightarrow{u}}\overrightarrow{v}$

a)

$\frac{18}{25}<3,0,4>$

b)

$\frac{18}{22}<3,0,4>$

c)

$\frac{18}{22}<2,3,3>$

d)

$\frac{18}{25}<2,3,3>$

81.

Given $\overrightarrow{u}=<1,-2>,\ \overrightarrow{v}=<-4,-3>$ , find $scal_{\overrightarrow{u}}\overrightarrow{v}$

a)

$\frac{2}{5}$

b)

$2$

c)

$\frac{2}{\sqrt[]{5}}$

d)

$-2$

82.

Given $\overrightarrow{u}=<1,-2>,\ \overrightarrow{v}=<-4,-3>$ , find $\operatorname{proj}_{\overrightarrow{v}}\overrightarrow{u}$

a)

$\frac{2}{25}<-4,-3>$

b)

$\frac{2}{25}<1,-2>$

c)

$\frac{2}{5}<-4,-3>$

d)

$\frac{2}{5}<1,-2>$

83.

Given $\overrightarrow{u}=<1,-2>,\ \overrightarrow{v}=<-4,-3>$ , find $\operatorname{proj}_{\overrightarrow{u}}\overrightarrow{v}$

a)

$\frac{2}{25}<-4,-3>$

b)

$\frac{2}{25}<1,-2>$

c)

$\frac{2}{5}<-4,-3>$

d)

$\frac{2}{5}<1,-2>$

84.

Calculate the cross product of <1, -2, 1> and <2, -1,1>

a)

5

b)

3

c)

< -1, -1, 3>

d)

< -1, 1, 3>

85.

Given parallel vectors u and v. Answer ALL that apply

a)

their dot product is zero

b)

u is a scalar multiple of v

c)

magnitude of u equals the magnitude of v

d)

their cross product is 0

86.

$\left(j+7k\right)\times\left(i+4j+5k\right)$

a)

-23i+7j-k

b)

23i+7j-k

c)

-2i+j-7k

d)

-3i+7j-k

87.

Find the cross product of <3,4,7> and <4,9,2>.

a)

<-55, 22, 11>

b)

<-55, -22, -9>>

c)

<71,34, 63>

d)

<-71, -34, -63>

88.

What is the area of the parallelogram spanned by the vectors a=<3,−3,1> and b=<4,9,2>?

a)

56

b)

$5\sqrt{70}$

c)

$5\sqrt{62}$

d)

$\sqrt{454}$

89.

The cross product (without introducing a scalar to the formula) of two vectors can be used to find which of the following?

a)

The area of a triangle

b)

The area of a circle

c)

The area of a parallelogram

d)

The area of a trapezoid

90.

What is the area of a parallelogram with 2 adjacent sides that are the vectors $2i+3j+k$ and $3i+2j+k$ ?

a)

$\sqrt[]{6}$

b)

$\sqrt[]{14}$

c)

$\sqrt[]{27}$

d)

$\sqrt[]{30}$

91.

What is the area of a triangle with two sides that are vectors $2i+3j+k$ and $6i+9j+3k$ ?

a)

$\frac{\sqrt[]{6}}{2}$

b)

$\frac{\sqrt[]{27}}{2}$

c)

$0$

d)

$\frac{\sqrt[]{30}}{2}$

92.

For any vectors $u$ and $v$ in $R^3$ , then $u\ \times v\ =\ v\ \times u$

a)

True

b)

False

93.

For any vectors $u$ and $v$ in $R^3$ , then $\left|u\ \times v\right|\ =\ \left|v\ \times u\right|$

a)

True

b)

False

94.

Find the area of the parallelogram formed by the vectors u = < 3, 0, 1> and v = < 1, -1, 2 >.

a)

$\sqrt{35}$

b)

5

c)

$\sqrt{34}$

d)

$\sqrt[]{10}$

95.

Find a vector that is orthogonal to the vectors u = -4i + j + 8k and v = 3i - 4j - 3k.

a)

<29, 12, 13>

b)

<29, -12, -13>

c)

<29, 12, -13>

d)

<29, -12, 13>

96.

Find the area of the triangle that has 2 sides that are the vectors u = <-1, 3, 5> and v = <2, -6, -3>

a)

$\frac{\sqrt[]{490}}{2}$

b)

$\frac{\sqrt[]{1570}}{2}$

c)

$\frac{35}{2}$

d)

$\frac{\sqrt[]{613}}{2}$

97.

Find the cross product of u and v.

u = <3, -6, 2>, v = <1, 5, -8>

a)

<38, 26, 21>

b)

<38, 26, 9>

c)

<1, 5, -8>

d)

<38, -26, 21>

98.

Consider points A(3,−1,2),B(2,1,5), and C(1,−2,−2). Find the area of the triangle formed by these 3 points.

a)

$\frac{5\sqrt[]{6}}{2}$

b)

$\frac{9\sqrt[]{3}}{2}$

c)

$7\sqrt[]{6}$

d)

$4\sqrt[]{6}$

99.

Which of the following could be used to calculate work done given a vector d for the displacement of an object and a vector F for the force applied to the object as well as the angle θ the force is applied compared to the horizontal?

a)

$\left|F\right|\left|d\right|\sin\theta$

b)

$\left|F\right|\left|d\right|\cos\theta$

c)

$F\cdot d$

d)

$\left|d\times F\right|$

100.

Which of the following could be used to calculate the magnitude of the torque applied given a vector r for the radius of the wrench and a vector F for the force applied to the wrench as well as the angle θ the force is applied compared to the horizontal?

a)

$\left|F\right|\left|r\right|\sin\theta$

b)

$\left|F\right|\left|r\right|\cos\theta$

c)

$F\cdot d$

d)

$\left|r\times F\right|$

Sections 13-1 to 13-4 Review

Total questions: 100

Worksheet time: 3hrs 20mins

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