Deck 12: Vectors and the Geometry of Space

Question 1

Compute ||-4a - 3b||.

A)
B)
C)
D)

Answer 1

Question 2

Compute -4a - 5b.

A)
B)
C)
D)

Answer 2

Question 3

Find a vector with the given magnitude in the same direction as the given vector. magnitude 6, v = i - 5j

A) 6i - 30j
B)
C)
D) 6i

Answer 3

Question 4

Determine whether the vectors a and b are parallel.

A) parallel
B) not parallel

Question 5

Find a unit vector in the same direction as the given vector. 10i - 2j

A)
B)
C)
D)

Question 6

Write the given vector as the product of its magnitude and a unit vector. (Write the unit vector so that it is in the same direction as the given vector. This means that the signs of the components of the unit vector should be the same as the signs of the corresponding components of the given vector.) From to

A)
B)
C)
D)

Question 7

Compute 2a - 3b.

A)
B)
C)
D)

Question 8

The motor of a small boat produces a speed of 2 mph in still water. The boat is travelling in flowing water with a current velocity given by . In what direction should the boat head to travel due east? (Represent the direction with a unit vector.)

A) i
B)
C)
D) The boat cannot travel due east against this current.

Question 9

Compute a + b.

A)
B)
C)
D)

Question 10

Illustrate the difference a - b graphically. a = -i + 4j; b = -3i + 3j

A)
B)
C)
D)

Question 11

Find the vector with the given polar form.

A)
B)
C)
D)

Question 12

Find two unit vectors parallel to the given vector. From to

A)
B)
C)
D)

Question 13

Write the given vector as the product of its magnitude and a unit vector. (Write the unit vector so that it is in the same direction as the given vector. This means that the signs of the components of the unit vector should be the same as the signs of the corresponding components of the given vector.)

A)
B)
C)
D)

Question 14

Illustrate the sum a + b graphically. a = -2i - j; b = -3i + 2j

A)
B)
C)
D)

Question 15

Compute ||-3a - 2b||.

A)
B)
C)
D)

Question 16

Suppose that there are two forces acting on a barge being towed along a river. One force is exerted due south by the river current and has a magnitude of 7 units (the exact nature of the force unit is unimportant). The other force is exerted by a tugboat and has a magnitude of 9 units towards the north and 3 units towards the east. What is the net force acting on the barge?

A) 9 units towards the north, 4 units towards the east
B) 16 units towards the north, 3 units towards the east
C) 2 units towards the north, 3 units towards the east
D) 3 units towards the north, 2 units towards the east

Question 17

Find the distance between the given points. (5, -7, 7), (7, 7, 7)

A)
B) 200
C)
D) 340

Question 18

Determine whether the vectors a and b are parallel.

A) parallel
B) not parallel

Question 19

Find two unit vectors parallel to the given vector.

A)
B)
C)
D)

Question 20

Find the vector with initial point A and terminal point B. A = (5, -4), B = (9, 5)

A)
B)
C)
D)

Question 21

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 2,

A)
B)
C)
D)

Question 22

Identify the plane as parallel to the xy-plane, xz-plane, or yz-plane. y = -6

A) xz-plane
B) xy-plane
C) yz-plane
D) xy-plane and yz-plane

Question 23

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 6,

A)
B)
C)
D)

Question 24

Use vectors to determine if the points (5, 30, 7), (3, 3, -2), (-6, 0, 9), and (-4, 27, 18) form a square.

A) yes
B) no

Question 25

Find comp_b a for the given vectors.

A)
B)
C)
D)

Question 26

Find proj_b a for the given vectors.

A)
B)
C)
D)

Question 27

Find a vector perpendicular to the given vector. Show all your work.

Question 28

Compute .

A)
B) -25
C)
D) 643

Question 29

A small communications satellite is being maneuvered by two on-board thrusters. One thruster generates a force with magnitude 12 newtons in a direction . In response to the applied thrusts, the satellite moves under a total force of (the components are in units of newtons). Find the thrust vector for the second thruster.

A)
B)
C)
D)

Question 30

Find the displacement vectors and and determine whether the points P = (8, 0, 4), Q = (0, -8, -4), and R = (5, -3, 1) are collinear (on the same line).

A) collinear
B) not collinear

Question 31

Determine if the vectors are orthogonal.

A) orthogonal
B) not orthogonal

Question 32

Find proj_b a for the given vectors.

A)
B)
C)
D)

Question 33

Compute .

A)
B) 64
C)
D) 2878

Question 34

Find an equation of the sphere with radius 11 and center (2, -4, 4).

A)
B)
C)
D)

Question 35

A constant force of pounds moves an object in a straight line from the point (0, 0) to the point (20, -19). Compute the work done.

A) -880
B) 0
C) -750
D) 1150

Question 36

Compute the angle between the vectors. Round your answer to two decimal places.

A)
B)
C)
D)

Question 37

Determine if the vectors are orthogonal.

A) orthogonal
B) not orthogonal

Question 38

Compute ||-2a - 3b||.

A)
B)
C)
D)

Question 39

Find comp_b a for the given vectors.

A)
B)
C)
D)

Question 40

Find a vector perpendicular to the given vector. Show all your work.

Question 41

If find a vector such that

A)
B)
C)
D)

Question 42

Find the indicated area. Area of the parallelogram with two adjacent sides formed by and

A)
B) 45
C) 81
D)

Question 43

Compute the given determinant.

A) -24
B) 4
C) 24
D) -4

Question 44

Find two unit vectors orthogonal to the two given vectors.

A)
B)
C)
D)

Question 45

Use the cross product to determine the angle betweeen the vectors, assuming that . Round to the nearest thousandth.

A) 1.571
B) 0.467
C) 0.735
D) 0.835

Question 46

Find two unit vectors orthogonal to the two given vectors.

A)
B)
C)
D)

Question 47

Prove that proj_c(a + b) = proj_c a + proj_c b for any non-zero vectors a, b, and c.

Question 48

Use the cross product to determine the angle between the vectors, assuming that . Round to the nearest thousandth.

A) 1.571
B) 0.149
C) 0.396
D) 1.175

Question 49

Compute the cross product a × b.

A)
B)
C)
D)

Question 50

Which of the following statements are true? i. ii. iii.

A) i only
B) ii only
C) i and ii only
D) i and iii only
E) iii only

Question 51

Assume that the ball is moving into the page (and away from you) with the indicated spin. Determine the direction of the Magnus force.

A) to the right
B) to the left
C) up
D) down
E) The Magnus force is zero.

Question 52

Find the indicated area. Area of the triangle with vertices , , and

A)
B)
C)
D)

Question 53

Assume that the ball is moving into the page (and away from you) with the indicated spin. Determine the direction of the Magnus force.

A) to the right
B) to the left
C) up
D) down
E) The Magnus force is zero.

Question 54

A constant force of pounds moves an object in a straight line a distance of 15 feet, and the work done is 110 ft-lb. If the motion of the object started at the point (0, 0), find the coordinates of the final position of the object. [Assume that the final position is somewhere in the first quadrant of the coordinate system.]

A)
B)
C)
D) The final position cannot be determined.

Question 55

Compute the cross product a × b.

A)
B)
C)
D)

Question 56

Use the Cauchy-Schwartz Inequality in n dimensions to show that ,
where m is an odd natural number.

Question 57

Who is doing more work: a weight lifter who is holding a 450-pound barbell motionless over his head, or senior citizen sitting on a park bench? Explain.

Question 58

If you apply a force of magnitude 39 pounds at the end of an 9-inch wrench at an angle of to the wrench, find the magnitude of the torque applied to the bolt. Round to the nearest tenth of an inch-pound.

A) 25.3 in-lb
B) 175.5 in-lb
C) 14.6 in-lb
D) 304.0 in-lb

Question 59

Find the indicated volume. Volume of the parallelpiped with three adjacent edges formed by , , and

A) 232
B)
C)
D) 312

Question 60

Find the distance from the point Q to the given line. Round to the nearest thousandth. , line through and

A) 17.117
B) 1.018
C) 17.647
D) 14.314

Question 61

Find parametric equations of the line through parallel to .

Question 62

Find symmetric equations of the line through and normal to the plane .

A)
B)
C)
D)

Question 63

Find symmetric equations of the line through and .

A)
B)
C)
D)

Question 64

Use the parallelpiped volume formula to determine if the vectors are coplanar.

A) coplanar
B) not coplanar

Question 65

Determine if the lines are parallel, skew or intersect.

A) parallel
B) skew
C) intersect

Question 66

Use geometry to identify the cross product.

A) 0
B) j
C) k
D) -i

Question 67

Find symmetric equations of the line through and parallel to .

A)
B)
C)
D)

Question 68

Determine if the lines are parallel, skew or intersect.

A) parallel
B) skew
C) intersect

Question 69

Find an equation of the given plane. The plane containing the point with normal vector

A) 3x + y + 7z = -22
B) 2x + 7y - 5z = 0
C) 2x + 7y - 5z = -22
D) 2x + 7y - 5z = 11

Question 70

Use the parallelpiped volume formula to determine if the vectors are coplanar.

A) coplanar
B) not coplanar

Question 71

Determine if the lines are parallel, skew or intersect.

A) parallel
B) skew
C) intersect

Question 72

Sketch the given plane. y + 2z = 6

A)
B)
C)
D)

Question 73

Sketch the given plane. 3x - y + 2z = 6

A)
B)
C)
D)

Question 74

In many farmers' barns and in many mechanics' shops, you can find a stout steel pipe about 2 to 3 feet in length with one end pinched so that the pinched end fits snugly over a wrench handle. What might explain the common presence of this object in these settings? Explain your answer in terms of the vector concepts you have learned.

Question 75

Find an equation of the given plane. The plane containing the points , and

A) 3x - 9y - 7z = 82
B) 9x - 13y - 10z = 137
C) 9x - 13y - 10z = 0
D) 58x + 64y - 31z = -185

Question 76

Find an equation of the given plane. The plane containing the point and perpendicular to the planes 7x + y - z = -5 and x - 6y - 4z = -3

A) -10x + 27y - 43z = -9
B) -10x + 27y - 43z = -78
C) 7x + y - z = 31
D) -10x + 27y - 43z = 416

Question 77

Show that . Show all of your work.

Question 78

Find parametric equations of the line through and perpendicular to both and .

Question 79

Find parametric equations of the line through and .

Question 80

Use geometry to identify the cross product.

A) -4j
B) 0
C) 4i
D) 4k