Deck 10: Vectors and the Geometry of Space

Question 1

A ball is thrown at an angle of to the ground. If the ball lands m away, what was the initial speed of the ball? Let .

A)
B)
C)
D)
E)

Answer 1

Question 2

The curvature of the curve given by the vector function is Use the formula to find the curvature of at the point .

A)
B)
C)
D)
E)

Answer 2

Question 3

The position function of a particle is given by When is the speed a minimum?

A)
B)
C)
D)
E)

Answer 3

Question 4

Find the velocity, acceleration, and speed of an object with position function for Sketch the path of the object and its velocity and acceleration vectors.

Question 5

Find the length of the curve

A)
B)
C)
D)

Question 6

Find the scalar tangential and normal components of acceleration of a particle with position vector

A)
B)
C)
D)

Question 7

Find the velocity of a particle that has the given acceleration and the given initial velocity.

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B)
C)
D)
E)

Question 8

For the curve given by , find the unit normal vector.

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

Question 9

Find the velocity, acceleration, and speed of an object with position function for Sketch the path of the object and its velocity and acceleration vectors.

Question 10

Find the velocity, acceleration, and speed of an object with position function for Sketch the path of the object and its velocity and acceleration vectors.

Question 11

Find the velocity and position vectors of an object with acceleration initial velocity and initial position

Question 12

A projectile is fired from ground level with an initial speed of 1100 ft/sec and an angle of elevation of
a. Find the range of the projectile.
b. What is the maximm height attained by the projectile?
c. What is the speed of the projectile at impact?
Round your answers to the nearest integer.

Question 13

Find the acceleration of a particle with the following position function.

A)
B)
C)
D)
E)

Question 14

Find the length of the curve

A)
B)
C)
D)

Question 15

A force with magnitude N acts directly upward from the xy-plane on an object with mass kg. The object starts at the origin with initial velocity . Find its position function.

A)

B)
C)
D)
E)

Question 16

Find the acceleration of a particle with the given position function.

A)
B)
C)
D)
E)

Question 17

Find the curvature of the curve .

A)11
B)
C)1
D)0

Question 18

Find the speed of a particle with the given position function.

A)
B)
C)
D)
E)

Question 19

Find the scalar tangential and normal components of acceleration of a particle with position vector

Question 20

Reparametrize the curve with respect to arc length measured from the point where in the direction of increasing .

A)
B)
C)
D)
E)

Question 21

Find a vector function that represents the curve of intersection of the two surfaces: The circular cylinder and the parabolic cylinder .

A)
B)
C)
D)
E)

Question 22

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 23

Identify the planes that are perpendicular.

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B)
C)
D)
E)

Question 24

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 yz-plane.

A)
B)
C)
D)
E)

Question 25

Find an equation for the surface consisting of all points that are equidistant from the point and the plane .

A)
B)
C)
D)
E)

Question 26

Find the derivative of the vector function.

A)
B)
C)
D)
E)

Question 27

Find the distance between the point and the plane Round your answer to two decimal place.

A)
B)
C)
D)
E)

Question 28

Find an equation of the set of all points equidistant from the points and

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B)
C)
D)
E)

Question 29

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

Question 30

Let .Find the domain of .

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B)
C)
D)
E)

Question 31

Find an equation of the plane with x-intercept , y-intercept and z-intercept .

A)
B)
C)
D)
E)

Question 32

Find the curvature of .

A)
B)
C)
D)
E)

Question 33

Find the point at which the line given by the parametric equations below intersects the plane.

A)
B)
C)
D)
E)

Question 34

Which of the given lines is parallel to the line

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B)
C)
D)
E)

Question 35

Find the curvature of the curve .

A)
B)
C)
D)

Question 36

Find the derivative of the vector function.

A)
B)
C)
D)
E)

Question 37

Find the derivative of the vector function.

A)
B)
C)
D)
E)

Question 38

At what point on the curve is the normal plane parallel to the plane ?

A)
B)
C)
D)
E)

Question 39

Classify the surface.

A)A circular paraboloid with vertex and axis the z-axis.
B)A hyperboloid of one sheet with center and axis parallel to the z-axis.
C)A cone with axis parallel to the z-axis and vertex .

Question 40

Reduce the equation to one of the standard forms.

A)
B)
C)
D)
E)

Question 41

Find .

A)
B)
C)
D)
E)

Question 42

Find parametric equations for the line through and

Question 43

Find .

A)

B) ,

C)

D)

E)

Question 44

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

Question 45

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

Question 46

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

Question 47

Find .

A)
B)
C)
D)
E)

Question 48

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 49

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 50

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

Question 51

Find the cross product .

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B)
C)
D)
E)

Question 52

Find .

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B)
C)
D)
E)

Question 53

Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as shown. Let ft, ft, , lb.

Question 54

Find correct to three decimal places where , , .

A)
B)
C)
D)
E)

Question 55

Find the unit vectors that are parallel to the tangent line to the curve at the point .

A)
B)
C)
D)
E)

Question 56

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

Question 57

Calculate the angle between a and b (correct to the nearest degree).

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

Question 58

Find .

A)
B)
C)27
D)90
E)

Question 59

Find a nonzero vector orthogonal to the plane through the points P, Q, and R.

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

Question 60

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 61

Find the center and the radius of the sphere that has the given equation. + + - 6x + 4y = 0

A)( 3, -2, 0), 13
B)( -3, 2, 0),
C)( 3, -2, 0),
D)( -3, 2, 0), 13

Question 62

Find the standard equation of the sphere with center C and radius r.C (3, -5, 3); r = 7

A)(x - 3)² + (y + 5)² + (z - 3)² = 49
B)(x - 3)² + (y + 5)² + (z - 3)² = 7
C)(x + 3)² + (y - 5)² + (z + 3)² = 49
D)(x + 3)² + (y - 5)² + (z + 3)² = 7

Question 63

Find the distance from to the xy-planes.

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

Question 64

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, 1, , isosceles
C)1, 2, , neither
D)1, 1, , both

Question 65

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

Question 66

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

Question 67

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

Question 68

Find an equation of the sphere with center that touches the xy-plane.

A)
B)
C)
D)
E)