Deck 11: Vectors and the Geometry of Space

Question 1

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)

Answer 1

C

Question 2

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

A)
B)
C)
D)

Answer 2

C

Question 3

Find the vector with initial point A and terminal point B. A = (4, 2), B = (-3, -7)

A)
B)
C)
D)

Answer 3

D

Question 4

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

A)
B)
C)
D)

Question 5

Compute -a - 2b.

A)
B)
C)
D)

Question 6

The motor of a small boat produces a speed of 3 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 7

Compute a + b.

A)
B)
C)
D)

Question 8

Determine whether the vectors a and b are parallel.

A) parallel
B) not parallel

Question 9

Compute ||2a - 3b||.

A)
B)
C)
D)

Question 10

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 11

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

A)
B)
C)
D)

Question 12

Find the distance between the given points. (0, -3, -1), (3, 6, 8)

A)
B) 171
C)
D) 67

Question 13

Find a unit vector in the same direction as the given vector. -2i + 5j

A)
B)
C)
D)

Question 14

Find two unit vectors parallel to the given vector.

A)
B)
C)
D)

Question 15

Find the vector with the given polar form.

A)
B)
C)
D)

Question 16

Compute ||-2a - 4b||.

A)
B)
C)
D)

Question 17

Find a vector with the given magnitude in the same direction as the given vector. magnitude 2, v = 5i - j

A) 10i - 2j
B)
C)
D) 2i

Question 18

Compute 3a + 5b.

A)
B)
C)
D)

Question 19

Determine whether the vectors a and b are parallel.

A) parallel
B) not parallel

Question 20

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 11 units towards the north and 3 units towards the east. What is the net force acting on the barge?

A) 11 units towards the north, 4 units towards the east
B) 18 units towards the north, 3 units towards the east
C) 4 units towards the north, 3 units towards the east
D) 3 units towards the north, 4 units towards the east

Question 21

Compute ||2a - 4b||.

A)
B)
C)
D)

Question 22

Compute .

A)
B) -1
C)
D) 6529

Question 23

Determine if the vectors are orthogonal.

A) orthogonal
B) not orthogonal

Question 24

A small communications satellite is being maneuvered by two on-board thrusters. One thruster generates a force with magnitude 13 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 25

A constant force of pounds moves an object in a straight line from the point (0, 0) to the point (23, -15). Compute the work done.

A) -595
B) 0
C) -635
D) 865

Question 26

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

Question 27

Find the displacement vectors and and determine whether the points P = (-1, -5, 2), Q = (1, -3, 4), and R = (0, -4, 3) are collinear (on the same line).

A) collinear
B) not collinear

Question 28

Compute .

A)
B) 17
C)
D) 169

Question 29

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

A)
B)
C)
D)

Question 30

Use vectors to determine if the points (3, -1, 0), (3, -3, -1), (0, 1, -9), and (0, 3, -8) form a square.

A) yes
B) no

Question 31

Find comp_b a for the given vectors.

A)
B)
C)
D)

Question 32

Find proj_b a for the given vectors.

A)
B)
C)
D)

Question 33

Find an equation of the sphere with radius 2 and center (-2, 6, 8).

A)
B)
C)
D)

Question 34

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 13,

A)
B)
C)
D)

Question 35

Find a vector with the given magnitude and in the same direction as the given vector. Magnitude 7,

A)
B)
C)
D)

Question 36

Find comp_b a for the given vectors.

A)
B)
C)
D)

Question 37

Identify the plane as parallel to the xy-plane, xz-plane, or yz-plane. y = -3

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

Question 38

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

Question 39

Find proj_b a for the given vectors.

A)
B)
C)
D)

Question 40

Determine if the vectors are orthogonal.

A) orthogonal
B) not orthogonal

Question 41

Find two unit vectors orthogonal to the two given vectors.

A)
B)
C)
D)

Question 42

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

A) 1.571
B) 0.384
C) 0.658
D) 0.912

Question 43

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

A) 94
B)
C)
D) 134

Question 44

Compute the cross product a × b.

A)
B)
C)
D)

Question 45

A constant force of pounds moves an object in a straight line a distance of 15 feet, and the work done is 65 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 46

Compute the given determinant.

A) -570
B) 26
C) 570
D) -26

Question 47

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

Question 48

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

Question 49

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 50

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

A)
B)
C)
D)

Question 51

If find a vector such that

A)
B)
C)
D)

Question 52

Find two unit vectors orthogonal to the two given vectors.

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

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

A) 14.036
B) 0.644
C) 11.460
D) 12.992

Question 55

Compute the cross product a × b.

A)
B)
C)
D)

Question 56

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

A)
B) 7
C) 43
D)

Question 57

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

A) 1.571
B) 0.056
C) 0.239
D) 1.332

Question 58

If you apply a force of magnitude 33 pounds at the end of an 10-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) 23.8 in-lb
B) 165.0 in-lb
C) 13.8 in-lb
D) 285.8 in-lb

Question 59

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 60

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 61

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

A) 4x - 9y - 3z = 63
B) 3x - 18y - z = 106
C) 3x - 18y - z = 0
D) 86x + 17y - 48z = 335

Question 62

Find symmetric equations of the line through and parallel to .

A)
B)
C)
D)

Question 63

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

A)
B)
C)
D)

Question 64

Show that . Show all of your work.

Question 65

Find parametric equations of the line through and .

Question 66

Use geometry to identify the cross product.

A) -5j
B) 0
C) 5i
D) 5k

Question 67

Sketch the given plane. y + 2z = 6

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 and perpendicular to the planes 5x + 5y - 2z = -8 and -5x - y - 5z = -8

A) -27x + 35y + 20z = -9
B) -27x + 35y + 20z = -20
C) 5x + 5y - 2z = -52
D) -27x + 35y + 20z = 180

Question 70

Determine if the lines are parallel, skew or intersect.

A) parallel
B) skew
C) intersect

Question 71

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

A) coplanar
B) not coplanar

Question 72

Find symmetric equations of the line through and .

A)
B)
C)
D)

Question 73

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

A)
B)
C)
D)

Question 74

Find parametric equations of the line through parallel to .

Question 75

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

A) -5x - 3y + 9z = -30
B) 9x - 8y - z = 0
C) 9x - 8y - z = -30
D) 9x - 8y - z = 1

Question 76

Determine if the lines are parallel, skew or intersect.

A) parallel
B) skew
C) intersect

Question 77

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 78

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

Question 79

Use geometry to identify the cross product.

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

Question 80

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

A) coplanar
B) not coplanar