Deck 12: Vectors and the Geometry of Space

Question 1

Find parametric equations for the line through

Answer 1

Question 2

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.

Answer 2

4

Question 3

Find the point of intersection.

Answer 3

Question 4

Draw a rectangular box with the origin and as opposite vertices and with its faces parallel to the coordinate planes.Find the length of the diagonal of the box.

Question 5

The tension T at each end of the chain has magnitude and makes an angle with the horizontal.What is the weight of the chain? Round the result to the nearest hundredth.

Question 6

Velocities have both direction and magnitude and thus are vectors.The magnitude of a velocity vector is called speed.Suppose that a wind is blowing from the direction N W at a speed of km/h.(This means that the direction from which the wind blows is west of the northerly direction.) A pilot is steering a plane in the direction N E at an airspeed (speed in still air) of The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind.The ground speed of the plane is the magnitude of the resultant.Find the ground speed of the plane.Round the result to the nearest hundredth.

Question 7

Find an equation for the surface obtained by rotating the line about the x-axis.

Question 8

Find the length of the median of side AB of the triangle with vertices

Question 9

Find an equation of the plane through the origin and parallel to the plane

Question 10

Calculate the given quantities if

Question 11

Find an equation of the sphere that passes through the point and has center b.Find the curve in which this sphere intersects the

Question 12

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the -plane and units to the right of it.

Question 13

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes

Question 14

Write inequalities to describe the solid upper hemisphere of the sphere of radius centered at the origin.

Question 15

Find the equation of the line through and perpendicular to the plane

Question 16

Find an equation of the sphere that passes through the point and has center

Question 17

A bicycle pedal is pushed by a foot with a force of a-N as shown.The shaft of the pedal is long.Find the magnitude of the torque about P correct to two decimal places.

Question 18

Find parametric equations for the line through and parallel to the vector

Question 19

Find an equation of the plane that passes through the line of intersection of the planes and is perpendicular to the plane

Question 20

A woman walks due west on the deck of a ship at The ship is moving north at a speed of Find the speed of the woman relative to the surface of the water.Round the result to the nearest tenth.

Question 21

Find an equation for the surface obtained by rotating the line about the x-axis.

Question 22

Find the point of intersection.

Question 23

Find parametric equations for the line through and parallel to the vector

Question 24

Write inequalities to describe the solid upper hemisphere of the sphere of radius centered at the origin.

Question 25

a.Find an equation of the sphere that passes through the point and has center b.Find the curve in which this sphere intersects the xy-plane.

Question 26

Find the magnitude of the torque about P correct to two decimal places if a force is applied as shown.

Question 27

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes

Question 28

Find the equation of the line through and perpendicular to the plane

Question 29

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the to the right of it.

Question 30

Find an equation of the plane through the origin and parallel to the plane

Question 31

Draw a rectangular box with the origin and as opposite vertices and with its faces parallel to the coordinate planes.Find the length of the diagonal of the box.

Question 32

Calculate the given quantities if

Question 33

A bicycle pedal is pushed by a foot with a force of as shown.The shaft of the pedal is long.Find the magnitude of the torque about P correct to two decimal places.

Question 34

Find parametric equations for the line through

Question 35

Write an inequality to describe the region consisting of all points between (but not on) the spheres of radius entered at the origin.

Question 36

Find an equation of the sphere that passes through the point and has center

Question 37

Find an equation of the plane that passes through the line of intersection of the planes and is perpendicular to the plane

Question 38

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.

Question 39

Find the length of the median of side AB of the triangle with vertices

Question 40

A woman walks due west on the deck of a ship at mi/h.The ship is moving north at a speed of Find the speed of the woman relative to the surface of the water.Round the result to the nearest tenth.

Question 41

Select the correct Answer: for each question.
Find the midpoint of the line segment joining the given points.(4, -2, -3) and (6, -4, -1)

A)(5, -3, -2)
B)(-1, 1, -1)
C)(-1, -3, -1)
D)(-1, 1, -2)

Question 42

Select the correct Answer: for each question.
Find the unit vectors that are parallel to the tangent line to the curve at the point

A)
B)
C)
D)
E)

Question 43

Select the correct Answer: for each question.
Find the standard equation of the sphere with center C and radius r.

A)
B)
C)
D)

Question 44

Select the correct Answer: for each question.
Find the center and radius of the sphere.

A)
B)
C)
D)
E)none of these

Question 45

Select the correct Answer: for each question.
Suppose you start at the origin, move along the x-axis a distance of units in the positive direction, and then move downward a distance of units.What are the coordinates of your position?

A)
B)
C)
D)
E)

Question 46

Select the correct Answer: for each question.
Determine whether the given points are collinear.A (2, 3, 2), B (-4, 0, 5), and C (4, 4, 1)

A)Not collinear
B)Collinear

Question 47

Select the correct Answer: for each question.
Find an equation for the surface obtained by rotating the parabola about the y-axis.

A)
B)
C)
D)
E)

Question 48

Select the correct Answer: for each question.
Find the work done by a force that moves an object from the point to the point along a straight line.The distance is measured in meters and the force in newtons.

A)
B)
C)
D)
E)

Question 49

Select the correct Answer: for each question.
Find parametric equations for the line through the point that is parallel to the plane and perpendicular to the line

A)
B)
C)
D)
E)

Question 50

Select the correct Answer: for each question.
Let and let u be a vector with length that starts at the origin and rotates in the xy - plane.Find the maximum value of the length of the vector

A)
B)
C)
D)
E)

Question 51

Select the correct Answer: for each question.
Find correct to three decimal places where

A)
B)
C)
D)
E)

Question 52

Select the correct Answer: for each question.
Find the distance from to the xy-planes.

A)8
B)16
C)10
D)2
E)4

Question 53

Select the correct Answer: for each question.
Find the center and the radius of the sphere that has the given equation.

A)
B)( -2, 4, 0), 20
C)
D)( 2, -4, 0), 20

Question 54

Select the correct Answer: for each question.
Find the distance between the planes.

A)
B)
C)
D)
E)

Question 55

Select the correct Answer: for each question.
Find the cross product

A)
B)
C)
D)
E)

Question 56

Select the correct Answer: for each question.
Find the distance between the point and the plane Round yourAnswer to two decimal place.

A)
B)
C)
D)
E)

Question 57

Select the correct Answer: for each question.
Find, correct to the nearest degree, the angle between the planes.

A)
B)
C)
D)
E)none of these

Question 58

Select the correct Answer: for each question.
Find a nonzero vector orthogonal to the plane through the points P, Q, and R.

A)
B)
C)
D)
E)None of these

Question 59

Select the correct Answer: for each question.
Plot the given points in a three-dimensional coordinate system.

A)
B)
C)
D)

Question 60

Select the correct Answer: for each question.
Find an equation of the sphere with center that touches the xy-plane.

A)
B)
C)
D)
E)

Question 61

Select the correct Answer for each question.
Find the distance from to the xy-planes.

A)8
B)16
C)10
D)2
E)4

Question 62

Select the correct Answer for each question.
Find the midpoint of the line segment joining the given points.

A)
B)(-1, 1, -1)
C)
D)(-1, 1, -2)

Question 63

Select the correct Answer for each question.
Reduce the equation to one of the standard forms.

A)
B)
C)
D)
E)

Question 64

Select the correct Answer for each question.
Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is times the distance from P to the

A)
B)
C)
D)
E)

Question 65

Select the correct Answer for each question.
Find an equation of the sphere with center that touches the xy-plane.

A)
B)
C)
D)
E)

Question 66

Select the correct Answer for each question.
Find the cross product

A)
B)
C)
D)
E)

Question 67

Select the correct Answer for each question.
Find the distance between the planes.

A)
B)
C)
D)
E)

Question 68

Select the correct Answer for each question.
Suppose you start at the origin, move along the x-axis a distance of units in the positive direction, and then move downward a distance of units.What are the coordinates of your position?

A)
B)
C)
D)
E)

Question 69

Select the correct Answer for each question.
Sketch the plane in a three-dimensional space represented by the equation.

A)
B)
C)
D)

Question 70

Select the correct Answer for each question.
Which of the given lines is parallel to the line

A)
B)
C)
D)
E)

Question 71

Select the correct Answer for each question.
Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither.A (-1, 0, 1), B (1, 1, -1), C (1, 1, 1)

A)3, 2, right
B)1, 2, neither
C)1, 1, both
D)1, 1, isosceles

Question 72

Select the correct Answer for each question.
Find the work done by a force that moves an object from the point to the point along a straight line.The distance is measured in meters and the force in newtons.

A)
B)
C)
D)
E)

Question 73

Select the correct Answer for each question.
Find an equation of the set of all points equidistant from the points and

A)
B)
C)
D)
E)

Question 74

Select the correct Answer for each question.
Find the center and the radius of the sphere that has the given equation.

A)
B)( -2, 4, 0), 20
C)
D)( 2, -4, 0), 20

Question 75

Select the correct Answer for each question.
Find the distance between the point and the plane Round yourAnswer to two decimal place.

A)
B)
C)
D)
E)

Question 76

Select the correct Answer for each question.
Determine whether the given points are collinear.

A)Not collinear
B)Collinear

Question 77

Select the correct Answer for each question.
Let and let u be a vector with length that starts at the origin and rotates in the xy - plane.Find the maximum value of the length of the vector

A)
B)
C)
D)
E)

Question 78

Select the correct Answer for each question.
Find a nonzero vector orthogonal to the plane through the points P, Q, and R.

A)
B)
C)
D)
E)None of these

Question 79

Select the correct Answer for each question.
Find that the midpoint of the line segment from

A)
B)
C)
D)
E)

Question 80

Select the correct Answer for each question.
Find, correct to the nearest degree, the angle between the planes.

A)
B)
C)
D)
E)none of these