Question 1

Find the midpoint of the line segment joining the given points. (-5, 0, 2) and (-3, -2, 4)

Accepted Answer

A)  ( $- 1$ , $$- 1$$ , $- 1$ ) 
B)  ( $- 1$ , $1$ , $3$ ) 
C)  ( $- 1$ , $1$ , $- 1$ ) 
D)  ( $- 4$ , $- 1$ , $3$ ) 
A)  ( $- 1$ , $$- 1$$ , $- 1$ ) 
B)  ( $- 1$ , $1$ , $3$ ) 
C)  ( $- 1$ , $1$ , $- 1$ ) 
D)  ( $- 4$ , $- 1$ , $3$ )

Question 2

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?

Accepted Answer

A)  $( 6,1,0 )$ 
B)  $( 6,0 , - 1 )$ 
C)  $( 6,0,1 )$ 
D)  $( 6 , - 1,0 )$ 
E)  $( 0,6,1 )$ 
A)  $( 6,1,0 )$ 
B)  $( 6,0 , - 1 )$ 
C)  $( 6,0,1 )$ 
D)  $( 6 , - 1,0 )$ 
E)  $( 0,6,1 )$

Question 3

Identify the planes that are perpendicular.

Accepted Answer

A)  $10 x - y - z = 6 , - 9 x + y - 19 z = 2$ 
B)  $x - 10 y = 6 , - 9 x - 19 z = 2$ 
C)  $x + 10 y = 6 , - 9 x - y - 19 z = 2$ 
D)  $x + 10 y = 6 , - 9 x - 19 z = 2$ 
E)  $x + 10 y - z = 6 , - 9 x - y - 19 z = 2$ 
A)  $10 x - y - z = 6 , - 9 x + y - 19 z = 2$ 
B)  $x - 10 y = 6 , - 9 x - 19 z = 2$ 
C)  $x + 10 y = 6 , - 9 x - y - 19 z = 2$ 
D)  $x + 10 y = 6 , - 9 x - 19 z = 2$ 
E)  $x + 10 y - z = 6 , - 9 x - y - 19 z = 2$

Question 4

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.

Accepted Answer

The answer of Write an inequality to describe the half-space...

Question 5

Find the length of the median of side AB of the triangle with vertices $A ( - 8,8 , - 7 ) , B ( - 4 , - 8 , - 5 )$ $\text { and } C ( 10 , - 2,6 ) \text {. }$

Accepted Answer

The answer of Find the length of the median of...

Question 6

Find the point of intersection. $L _ { 1 } : \frac { x - 17 } { 3 } = \frac { y - 58 } { 8 } = \frac { z - 23 } { 2 } , L _ { 2 } : \frac { x - 49 } { 7 } = \frac { y - 26 } { 4 } = z - 15$

Accepted Answer

The answer of Find the point of intersection. \(L _...

Question 7

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

Accepted Answer

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 
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 8

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$ is perpendicular to the plane $x + y - 3 z = 2$

Accepted Answer

The answer of Find an equation of the plane that...

Question 9

Find an equation of the sphere that passes through the point $( 7 , - 8,9 )$ and has center $( - 8,9 , - 9 )$ .

Accepted Answer

The answer of Find an equation of the sphere that...

Question 10

Find an equation of the plane that passes through the point $( 1,0 , - 3 )$ and contains the line $x = 6 - 7 t , y = 6 + 5 t , \quad z = 4 + 8 t$

Accepted Answer

The answer of Find an equation of the plane that...

Question 11

Determine whether the given points are collinear. A (-3, -2, -3), B (-9, -5, 0), and C (-1, -1, -4)

Accepted Answer

A)  Not collinear 
B)  Collinear 
A)  Not collinear 
B)  Collinear

Question 12

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

Accepted Answer

The answer of Write inequalities to describe the solid upper...

Question 13

Two forces $F _ { 1 } \text { and } F _ { 2 }$ with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure. Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth.

Accepted Answer

A)  $$17.7 \mathrm { lb }$$ 
B)  $13.7 \mathrm { lb }$ 
C)  $15.7 \mathrm { lb }$ 
D)  $19.7 \mathrm { lb }$ 
E)  $11.7 \mathrm { lb }$ 
A)  $$17.7 \mathrm { lb }$$ 
B)  $13.7 \mathrm { lb }$ 
C)  $15.7 \mathrm { lb }$ 
D)  $19.7 \mathrm { lb }$ 
E)  $11.7 \mathrm { lb }$

Question 14

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$

Accepted Answer

A)  $\mathbf { i } - 7 \mathbf { j } + 8 \mathbf { k }$ 
B)  $- 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k }$ 
C)  $\mathbf { i } - 8 \mathbf { j } + 7 \mathbf { k }$ 
D)  $8 \mathbf { i } - 6 \mathbf { j } + 56 \mathbf { k }$ 
E)  None of these 
A)  $\mathbf { i } - 7 \mathbf { j } + 8 \mathbf { k }$ 
B)  $- 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k }$ 
C)  $\mathbf { i } - 8 \mathbf { j } + 7 \mathbf { k }$ 
D)  $8 \mathbf { i } - 6 \mathbf { j } + 56 \mathbf { k }$ 
E)  None of these

Question 15

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.

Accepted Answer

The answer of Write an inequality to describe the region...

Question 16

Sketch the plane in a three-dimensional space represented by the equation. z = 2

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 17

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

Accepted Answer

A)  $$\frac { x } { 11 } + \frac { y } { 5 } + \frac { z } { 5 } = 1$$ 
B)  $11 x + 5 y + 5 z = 0$ 
C)  $\frac { 5 x } { 5 } + 11 y + z = 0$ 
D)  $\frac { x } { 11 } + \frac { y } { 5 } + \frac { z } { 5 } = 1$ 
E)  $11 x + 5 y + 5 z = 1$ 
A)  $$\frac { x } { 11 } + \frac { y } { 5 } + \frac { z } { 5 } = 1$$ 
B)  $11 x + 5 y + 5 z = 0$ 
C)  $\frac { 5 x } { 5 } + 11 y + z = 0$ 
D)  $\frac { x } { 11 } + \frac { y } { 5 } + \frac { z } { 5 } = 1$ 
E)  $11 x + 5 y + 5 z = 1$

Question 18

Find parametric equations for the line through $( - 6,4,7 )$ and parallel to the vector $\{ 7,5 , - 7 \}$

Accepted Answer

The answer of Find parametric equations for the line through...

Question 19

Let $v = 7 j$ 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 $| \mathbf { u } \times \mathbf { v } |$ .

Accepted Answer

A)  $| \mathbf { u } \times \mathbf { v } | = 5$ 
B)  $| \mathbf { u } \times \mathbf { v } | = 7$ 
C)  $| \mathbf { u } \times \mathbf { v } | = 35$ 
D)  $| \mathbf { u } \times \mathbf { v } | = 157$ 
E)  $| \mathbf { u } \times \mathbf { v } | = 1$ 
A)  $| \mathbf { u } \times \mathbf { v } | = 5$ 
B)  $| \mathbf { u } \times \mathbf { v } | = 7$ 
C)  $| \mathbf { u } \times \mathbf { v } | = 35$ 
D)  $| \mathbf { u } \times \mathbf { v } | = 157$ 
E)  $| \mathbf { u } \times \mathbf { v } | = 1$

Question 20

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Accepted Answer

A)  $64 \mathrm {~J}$ 
B)  $74 \mathrm {~J}$ 
C)  $84 \mathrm {~J}$ 
D)  $94 \mathrm {~J}$ 
E)  $104 \mathrm {~J}$ 
A)  $64 \mathrm {~J}$ 
B)  $74 \mathrm {~J}$ 
C)  $84 \mathrm {~J}$ 
D)  $94 \mathrm {~J}$ 
E)  $104 \mathrm {~J}$

Find the midpoint of the line segment joining the given points. (-5, 0, 2) and (-3, -2, 4)

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?

Identify the planes that are perpendicular.

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.

Find the length of the median of side AB of the triangle with vertices $A ( - 8,8 , - 7 ) , B ( - 4 , - 8 , - 5 )$ $\text { and } C ( 10 , - 2,6 ) \text {. }$

Find the point of intersection. $L _ { 1 } : \frac { x - 17 } { 3 } = \frac { y - 58 } { 8 } = \frac { z - 23 } { 2 } , L _ { 2 } : \frac { x - 49 } { 7 } = \frac { y - 26 } { 4 } = z - 15$

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

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$ is perpendicular to the plane $x + y - 3 z = 2$

Find an equation of the sphere that passes through the point $( 7 , - 8,9 )$ and has center $( - 8,9 , - 9 )$ .

Find an equation of the plane that passes through the point $( 1,0 , - 3 )$ and contains the line $x = 6 - 7 t , y = 6 + 5 t , \quad z = 4 + 8 t$

Determine whether the given points are collinear. A (-3, -2, -3), B (-9, -5, 0), and C (-1, -1, -4)

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

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$

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.

Sketch the plane in a three-dimensional space represented by the equation. z = 2

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

Find parametric equations for the line through $( - 6,4,7 )$ and parallel to the vector $\{ 7,5 , - 7 \}$

Let $v = 7 j$ 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 $| \mathbf { u } \times \mathbf { v } |$ .

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 12: Vectors and the Geometry of Space

Find the midpoint of the line segment joining the given points. (-5, 0, 2) and (-3, -2, 4)

Suppose you start at the origin, move along the x-axis a distance of 666 units in the positive direction, and then move downward a distance of 111 units. What are the coordinates of your position?

Identify the planes that are perpendicular.

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xz-plane and 333 units to the right of it.

Find the length of the median of side AB of the triangle with vertices A(−8,8,−7),B(−4,−8,−5)A ( - 8,8 , - 7 ) , B ( - 4 , - 8 , - 5 )A(−8,8,−7),B(−4,−8,−5) and C(10,−2,6). \text { and } C ( 10 , - 2,6 ) \text {. } and C(10,−2,6).

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

Find an equation of the plane that passes through the line of intersection of the planes x−z=1x - z = 1x−z=1 and y+2z=3y + 2 z = 3y+2z=3 is perpendicular to the plane x+y−3z=2x + y - 3 z = 2x+y−3z=2

Find an equation of the sphere that passes through the point (7,−8,9)( 7 , - 8,9 )(7,−8,9) and has center (−8,9,−9)( - 8,9 , - 9 )(−8,9,−9) .

Find an equation of the plane that passes through the point (1,0,−3)( 1,0 , - 3 )(1,0,−3) and contains the line x=6−7t,y=6+5t,z=4+8tx = 6 - 7 t , y = 6 + 5 t , \quad z = 4 + 8 tx=6−7t,y=6+5t,z=4+8t

Determine whether the given points are collinear. A (-3, -2, -3), B (-9, -5, 0), and C (-1, -1, -4)

Write inequalities to describe the solid upper hemisphere of the sphere of radius 555 centered at the origin.

Two forces F1 and F2F _ { 1 } \text { and } F _ { 2 }F1​ and F2​ with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure. Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth.

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. P(1,0,0),Q(7,8,0),R(0,8,1)P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )P(1,0,0),Q(7,8,0),R(0,8,1)

Write an inequality to describe the region consisting of all points between (but not on) the spheres of radius 999 and 111111 centered at the origin.

Sketch the plane in a three-dimensional space represented by the equation. z = 2

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

Find parametric equations for the line through (−6,4,7)( - 6,4,7 )(−6,4,7) and parallel to the vector {7,5,−7}\{ 7,5 , - 7 \}{7,5,−7}

Let v=7jv = 7 jv=7j and let u be a vector with length 555 that starts at the origin and rotates in the xy - plane. Find the maximum value of the length of the vector ∣u×v∣| \mathbf { u } \times \mathbf { v } |∣u×v∣ .

Find the work done by a force F=i+5j+7k\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }F=i+5j+7k that moves an object from the point (3,0,5)( 3,0,5 )(3,0,5) to the point (7,5,10)( 7,5,10 )(7,5,10) along a straight line. The distance is measured in meters and the force in newtons.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

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?

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.

Find the length of the median of side AB of the triangle with vertices $A ( - 8,8 , - 7 ) , B ( - 4 , - 8 , - 5 )$ $\text { and } C ( 10 , - 2,6 ) \text {. }$

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$ is perpendicular to the plane $x + y - 3 z = 2$

Find an equation of the sphere that passes through the point $( 7 , - 8,9 )$ and has center $( - 8,9 , - 9 )$ .

Find an equation of the plane that passes through the point $( 1,0 , - 3 )$ and contains the line $x = 6 - 7 t , y = 6 + 5 t , \quad z = 4 + 8 t$

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

Find a nonzero vector orthogonal to the plane through the points P, Q, and R. $P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$

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.

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

Find parametric equations for the line through $( - 6,4,7 )$ and parallel to the vector $\{ 7,5 , - 7 \}$

Let $v = 7 j$ 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 $| \mathbf { u } \times \mathbf { v } |$ .

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.