Deck 10: Ectors and the Geometry of Space

Question 1

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

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

Reparametrize the curve with respect to arc length measured from the point where t = 0. in the direction of increasing t.

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B)
C)
D)
E)

Answer 3

Question 4

Find the length of the curve

A)
B)
C)
D)

Question 5

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 6

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 maximum height attained by the projectile?
c.What is the speed of the projectile at impact?
Round your answers to the nearest integer.

Question 7

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

A)
B)
C)
D)
E)

Question 8

Find the length of the curve

A)
B)
C)
D)

Question 9

Find the velocity,acceleration,and speed of an object with position function for t =1. Sketch the path of the object and its velocity and acceleration vectors.

Question 10

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

A)
B)
C)
D)
E)

Question 11

Find the curvature of the curve .

A) 0
B) 1
C) 11
D)

Question 12

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

A)
B)
C)
D)
E)

Question 13

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

Question 14

Find the velocity,acceleration,and speed of an object with position function for t = 1. Sketch the path of the object and its velocity and acceleration vectors.

Question 15

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

Question 16

A force with magnitude 12 N acts directly upward from the xy-plane on an object with mass 2 kg.The object starts at the origin with initial velocity .Find its position function.

A)
B)
C)
D)
E)

Question 17

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

A)
B)
C)
D)

Question 18

A ball is thrown at an angle of to the ground.If the ball lands 30 m away,what was the initial speed of the ball? Let .

A)
B)
C)
D)

Question 19

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

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

Question 20

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

A)
B)
C)
D)
E)

Question 21

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

A)
B)
C)
D)

Question 22

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

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B)
C)
D)
E)

Question 23

Find parametric equations for the line through the point (3,4,5) that is parallel to the plane and perpendicular to the line

A)
B)
C)
D)
E)

Question 24

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

Question 25

Let . Find the domain of .

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B)
C)
D)
E)

Question 26

Find the curvature of .

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B)
C)
D)
E)

Question 27

Reduce the equation to one of the standard forms.

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B)
C)
D)
E)

Question 28

Classify the surface.

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

Question 29

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 30

Find the derivative of the vector function.

A)
B)
C)
D)
E)

Question 31

Which of the given lines is parallel to the line

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B)
C)
D)
E)

Question 32

Find the derivative of the vector function.

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B)
C)
D)
E)

Question 33

Find the distance between the point (9,9,3) and the plane Round your answer to two decimal place.

A)
B)
C)
D)
E)

Question 34

Find an equation of the plane with x-intercept = 11 ,y-intercept = 5 and z-intercept = 5.

A)

B)
C)
D)

Question 35

Find the curvature of the curve .

A)
B)
C)
D)

Question 36

Identify the planes that are perpendicular.

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B)
C)
D)
E)

Question 37

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is three times the distance from P to the yz-plane.

A)
B)
C)
D)
E)

Question 38

Find the derivative of the vector function.

A)
B)
C)
D)
E)

Question 39

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 40

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

A)
B)
C)
D)
E)

Question 41

Find .

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

Question 42

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

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

Question 43

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

Question 44

Find .

A)
B)
C)
D)
E)

Question 45

Find correct to three decimal places where , , .

A)
B)
C)
D)

Question 46

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

Question 47

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

Question 48

Find the cross product .

A)
B)
C)
D)
E)

Question 49

A bicycle pedal is pushed by a foot with a force of a-N as shown.The shaft of the pedal is 12 cm long. Find the magnitude of the torque about P correct to two decimal places.

Question 50

The tension T at each end of the chain has magnitude 27 N and makes an angle with the horizontal.What is the weight of the chain? Round the result to the nearest hundredth.

Question 51

Find .

A)
B)
C) ,
D)
E)

Question 52

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 53

Let and let u be a vector with length 5 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 54

Suppose you start at the origin,move along the x-axis a distance of 6 units in the positive direction,and then move downward a distance of 1 units.What are the coordinates of your position?

A)
B)
C)
D)
E)

Question 55

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

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

Question 56

A woman walks due west on the deck of a ship at 6 mi/h.The ship is moving north at a speed of 20 mi/h.Find the speed of the woman relative to the surface of the water.Round the result to the nearest tenth.

Question 57

Find parametric equations for the line through and

Question 58

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

A)
B)
C)
D)
E)

Question 59

Find .

A)
B)
C)
D)
E)

Question 60

Find .

A)
B)
C)
D)
E)

Question 61

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

A)
B)
C)
D)
E)

Question 62

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xz-plane and 3 units to the right of it.

Question 63

Find the distance from to the xy-planes.

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

Question 64

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),13
D) ( -3,2,0),

Question 65

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

Question 66

Write an inequality to describe the region consisting of all points between (but not on) the spheres of radius 9 and 11 centered at the origin.

Question 67

Write inequalities to describe the solid upper hemisphere of the sphere of radius 5 centered at the origin.

Question 68

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) 1,1, ,both
D) 1,2, ,neither