Deck 9: Vectors in Two and Three Dimensions

Question 1

Find and , given that and .

Answer 1

,

Question 2

Given and in the figure, sketch .

Answer 2

Question 3

Find the vector with initial point and terminal point .

Answer 3

The vector is given by .

Question 4

Find the vector with initial point and terminal point .

Question 5

Find the horizontal and vertical components of the vector with length , and direction .

Question 6

A jet is flying in a direction N E with a speed of mi h, find the north and east components of the velocity.

Question 7

Find the magnitude and direction (in degrees) of the vector .

Question 8

Find the vector with initial point and terminal point .

Question 9

Find the magnitude and direction (in degrees) of the vector .

Question 10

Find the vector with initial point and terminal point .

Question 11

Find the magnitude and direction (in degrees) of the vector .

Question 12

Find the magnitude and direction (in degrees) of the vector .

Question 13

Find the magnitude and direction (in degrees) of the vector .

Question 14

Find the vector with initial point and terminal point .

Question 15

Find , and , given that , and .

Question 16

Find the vector with initial point and terminal point .

Question 17

Given and in the figure, sketch .

Question 18

Find and , given that and .

Question 19

Find the vector with initial point and terminal point .

Question 20

Given and in the figure, sketch .

Question 21

Find the component of along if and .

Question 22

Find and the angle between and for , .

Question 23

Find the work done by the force in moving an object from to .

Question 24

A man pulls a sled horizontally by exerting lb on the rope that is tied to its front end. If the rope makes an angle of with horizontal, find the work done in moving the sled ft.

Question 25

Determine whether is orthogonal to .

Question 26

Given and , calculate and then resolve into and .

Question 27

Given and , calculate and then resolve into and .

Question 28

Find and the angle between and for .

Question 29

Find and the angle between and for , .

Question 30

Find and the angle between and for .

Question 31

Determine whether is orthogonal to .

Question 32

Find and the angle between and for .

Question 33

Determine whether is orthogonal to .

Question 34

Given that the forces , and are acting on a point , find the resultant force, magnitude and the additional force required in order for the forces to be in equilibrium.

Question 35

Find the component of along if and .

Question 36

Find the work done by the force in moving an object from to .

Question 37

Given and , calculate and then resolve into and .

Question 38

Find and the angle between and for .

Question 39

Determine whether is orthogonal to .

Question 40

A jet heads in the direction N E at a speed of mi/h. If the jet experiences a mi/h crosswind flowing due east, find
(a) the true velocity
of the jet.
(b) the true speed of the jet.
(c) the direction of the jet.

Question 41

Describe the surface represented by the given equation.

Question 42

Describe the surface represented by the given equation.

Question 43

Find an equation of a sphere with the given radius r and center C. ;

Question 44

Describe the surface represented by the given equation.

Question 45

Find the distance between the points and .

Question 46

A car is on a ramp that is inclined to the horizontal. If the car weighs lb, find the force required to keep the car from rolling down the ramp.

Question 47

Find the center and radius of the sphere.

Question 48

A water tank is in the shape of a sphere of radius 10 feet. The tank is supported on a metal circle 8 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Question 49

Find the center and radius of the sphere.

Question 50

Find the distance between the points and .

Question 51

Find the center and radius of the sphere.

Question 52

Describe the surface represented by the given equation.

Question 53

Find the distance between the points and .

Question 54

Find an equation of a sphere with the given radius r and center C. ;

Question 55

Find the center and radius of the sphere.

Question 56

Find the distance between the points and .

Question 57

Find an equation of a sphere with the given radius r and center C. ;

Question 58

Describe the trace of the sphere in
(a) the yz-plane and
(b) the plane
.

Question 59

Find an equation of a sphere with the given radius r and center C. ;

Question 60

Describe the trace of the sphere through
(a) the xz-plane in
(b) the plane
.

Question 61

Express the given vector in terms of the unit vectors i, j, and k.

Question 62

Two vectors u and v are given. Find their dot product .

Question 63

Find the vectors , , and .

Question 64

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 0.12 , | \mathbf { b } | = 1.25 , \theta = 85 ^ { \circ }$

Question 65

Find the magnitude of the given vector.

Question 66

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 4 , |\mathbf { b }| = 5 , \theta = 60 ^ { \circ }$

Question 67

Find the vector v with initial point P and terminal point Q. ,

Question 68

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v.

Question 69

If the vector v has initial point P, what is its terminal point?

Question 70

Find a vector that is perpendicular to the plane passing through the three given points.

Question 71

Find the magnitude of the given vector.

Question 72

Find the vectors , , and .

Question 73

Express the given vector in terms of the unit vectors i, j, and k.

Question 74

Two vectors u and v are given. Find their dot product .

Question 75

For the given vectors a and b, find the cross product .

Question 76

If the vector v has initial point P, what is its terminal point?

Question 77

Find the vector v with initial point P and terminal point Q. ,

Question 78

For the given vectors a and b, find the cross product .

Question 79

A water tank is in the shape of a sphere of radius 15 feet. The tank is supported on a metal circle 12 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Question 80

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v.