Question 1

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

Accepted Answer

The answer of \[\text { Find an equation of the...

Question 2

Find that the midpoint of the line segment from $P 1 ( - 20,20,3 )$ to $P 2 ( 6,24,7 )$.

Accepted Answer

A)  $( - 7 , - 22 , - 5 )$ 
B)  $( - 7,5,22 )$ 
C)  $( - 7,22,5 )$ 
D)  $( - 7,22 , - 5 )$ 
E)  $( - 7 , - 22,5 )$ 
A)  $( - 7 , - 22 , - 5 )$ 
B)  $( - 7,5,22 )$ 
C)  $( - 7,22,5 )$ 
D)  $( - 7,22 , - 5 )$ 
E)  $( - 7 , - 22,5 )$

Question 3

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 $\mathrm { N } \mathrm {} 60 ^ { \circ } \mathrm { W }$ at a speed of 80 $\mathrm { km } / \mathrm { h }$. (This means that the direction from which the wind blows is $60 ^ { \circ }$ west of the northerly direction.) A pilot is steering a plane in the direction $\mathrm { N } \mathrm {} 45 ^ { \circ } \mathrm { E }$ at an airspeed (speed in still air) of $250 \mathrm {~km} / \mathrm { h }$. 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.

Accepted Answer

The answer of Velocities have both direction and magnitude and...

Question 4

Find the point at which the line given by the parametric equations below intersects the plane.
$$2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t$$
Select the correct answer.

Accepted Answer

A)  $( 38 , - 10 , - 28 )$ 
B)  $( - 43 , - 10,18 )$ 
C)  $( - 38 , - 10,28 )$ 
D)  $( 43 , - 10 , - 18 )$ 
E)  $( 38 , - 10,28 )$ 
A)  $( 38 , - 10 , - 28 )$ 
B)  $( - 43 , - 10,18 )$ 
C)  $( - 38 , - 10,28 )$ 
D)  $( 43 , - 10 , - 18 )$ 
E)  $( 38 , - 10,28 )$

Question 5

Find the volume of the parallelepiped with adjacent edges $P Q , P R$, and $P S$.
$$P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )$$

Accepted Answer

The answer of Find the volume of the parallelepiped with...

Question 6

Find, correct to the nearest degree, the angle between the planes.
$$x + y - z = 1,8 x - 11 y + 16 z = 5$$

Accepted Answer

The answer of Find, correct to the nearest degree, the...

Question 7

Identify the planes that are perpendicular. Select the correct answer.

Accepted Answer

A)  $x + 8 y - z = 6 , - 5 x - y - 13 z = 2$ 
B)  $x + 8 y = 6 , - 5 x - y - 13 z = 2$ 
C)  $x - 8 y = 6 , - 5 x - 13 z = 2$ 
D)  $x + 8 y = 6 , - 5 x - 13 z = 2$ 
E)  $8 x - y - z = 6 , - 5 x + y - 13 z = 2$ 
A)  $x + 8 y - z = 6 , - 5 x - y - 13 z = 2$ 
B)  $x + 8 y = 6 , - 5 x - y - 13 z = 2$ 
C)  $x - 8 y = 6 , - 5 x - 13 z = 2$ 
D)  $x + 8 y = 6 , - 5 x - 13 z = 2$ 
E)  $8 x - y - z = 6 , - 5 x + y - 13 z = 2$

Question 8

Find the center and radius of the sphere. Select the correct answer.
$$x ^ { 2 } - 4 x - 24 + y ^ { 2 } + 16 y + z ^ { 2 } - 12 z = 0$$

Accepted Answer

A)  $C ( - 1 , - 8 , - 9 ) , r = \sqrt { 2 }$ 
B)  $C ( 1 , - 8 , - 9 ) , r = 4$ 
C)  $C ( - 1 , - 8 , - 9 ) , r = 4 \sqrt { 2 }$ 
D)  $C ( - 1 , - 8 , - 9 ) , r = 4$ 
E)  none of these 
A)  $C ( - 1 , - 8 , - 9 ) , r = \sqrt { 2 }$ 
B)  $C ( 1 , - 8 , - 9 ) , r = 4$ 
C)  $C ( - 1 , - 8 , - 9 ) , r = 4 \sqrt { 2 }$ 
D)  $C ( - 1 , - 8 , - 9 ) , r = 4$ 
E)  none of these

Question 9

Classify the surface.
$$23 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 12 y + 12 z = 0$$

Accepted Answer

A cone with axis par

Question 10

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes $x = 1 , y = 8$, and $z = 4$.

Accepted Answer

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

Question 11

$$\text { Find an equation of the sphere that passes through the point } ( 7 , - 4,9 ) \text { and has center } ( - 4,5 , - 5 ) \text {. }$$

Accepted Answer

The answer of \[\text { Find an equation of the...

Question 12

Find parametric equations for the line through the point $( 5,6,3 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$.
Select the correct answer.

Accepted Answer

A)  $\quad x = 4 t + 5 , y = - 2 t + 6 , z = - 2 t + 3$ 
B)  $x = 4 t , y = - 2 t + 6 , z = - 2 t$ 
C)  $x = 4 t + 2 , y = - 2 t + 6 , z = - 2 t + 3$ 
D)  $x = 4 t + 2 , y = - 2 t + 6 , z = 0$ 
E)  $x = 4 t , y = - 2 t , z = - 2 t + 3$ 
A)  $\quad x = 4 t + 5 , y = - 2 t + 6 , z = - 2 t + 3$ 
B)  $x = 4 t , y = - 2 t + 6 , z = - 2 t$ 
C)  $x = 4 t + 2 , y = - 2 t + 6 , z = - 2 t + 3$ 
D)  $x = 4 t + 2 , y = - 2 t + 6 , z = 0$ 
E)  $x = 4 t , y = - 2 t , z = - 2 t + 3$

Question 13

$$\text { Find an equation for the surface obtained by rotating the parabola } y = 3 x ^ { 2 } \text { about the } y \text {-axis. }$$

Accepted Answer

The answer of \[\text { Find an equation for the...

Question 14

Sketch the plane in a three-dimensional space represented by the equation.
$$z = - 1$$

Accepted Answer

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

Question 15

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 16

Find parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$.

Accepted Answer

A)  $x = 4 t + 2 , y = - 2 t + 4 , z = - 2 t + 5$ 
B)  $x = 4 t + 2 , y = - 2 t + 4 , z = 0$ 
C)  $x = 4 t , y = - 2 t , z = - 2 t + 5$ 
D)  $x = 4 t , y = - 2 t + 4 , z = - 2 t$ 
E)  $x = 4 t + 3 , y = - 2 t + 4 , z = - 2 t + 5$ 
A)  $x = 4 t + 2 , y = - 2 t + 4 , z = - 2 t + 5$ 
B)  $x = 4 t + 2 , y = - 2 t + 4 , z = 0$ 
C)  $x = 4 t , y = - 2 t , z = - 2 t + 5$ 
D)  $x = 4 t , y = - 2 t + 4 , z = - 2 t$ 
E)  $x = 4 t + 3 , y = - 2 t + 4 , z = - 2 t + 5$

Question 17

$$\text { Find parametric equations for the line through } ( - 4,1,1 ) \text { and } ( 4,5,6 ) \text {. }$$

Accepted Answer

The answer of \[\text { Find parametric equations for the...

Question 18

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$-plane and 11 units to the right of it.

Accepted Answer

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

Question 19

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$-plane and 11 units to the right of it.

Accepted Answer

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

Question 20

Find the distance (correct to two decimal places) between the given parallel planes.
$$6 x + 7 y - 2 z = 12 \text { and } 12 x + 14 y - 2 z = 70$$

Accepted Answer

The answer of Find the distance (correct to two decimal...

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

Find that the midpoint of the line segment from $P 1 ( - 20,20,3 )$ to $P 2 ( 6,24,7 )$ .

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t$ Select the correct answer.

Find the volume of the parallelepiped with adjacent edges $P Q , P R$ , and $P S$ . $P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )$

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,8 x - 11 y + 16 z = 5$

Identify the planes that are perpendicular. Select the correct answer.

Find the center and radius of the sphere. Select the correct answer. $x ^ { 2 } - 4 x - 24 + y ^ { 2 } + 16 y + z ^ { 2 } - 12 z = 0$

Classify the surface. $23 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 12 y + 12 z = 0$

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes $x = 1 , y = 8$ , and $z = 4$ .

$\text { Find an equation of the sphere that passes through the point } ( 7 , - 4,9 ) \text { and has center } ( - 4,5 , - 5 ) \text {. }$

Find parametric equations for the line through the point $( 5,6,3 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$ . Select the correct answer.

$\text { Find an equation for the surface obtained by rotating the parabola } y = 3 x ^ { 2 } \text { about the } y \text {-axis. }$

Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$

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 parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$ .

$\text { Find parametric equations for the line through } ( - 4,1,1 ) \text { and } ( 4,5,6 ) \text {. }$

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$ -plane and 11 units to the right of it.

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$ -plane and 11 units to the right of it.

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 12 x + 14 y - 2 z = 70$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

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Exam 12: Vectors and the Geometry of Space

Find that the midpoint of the line segment from P1(−20,20,3)P 1 ( - 20,20,3 )P1(−20,20,3) to P2(6,24,7)P 2 ( 6,24,7 )P2(6,24,7) .

Find the point at which the line given by the parametric equations below intersects the plane. 2x+4y−3z=−48;x=10+7t,y=−10,z=7t2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t2x+4y−3z=−48;x=10+7t,y=−10,z=7t Select the correct answer.

Find the volume of the parallelepiped with adjacent edges PQ,PRP Q , P RPQ,PR , and PSP SPS . P(2,3,4),Q(4,5,4),R(4,3,5),S(4,3,4)P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )P(2,3,4),Q(4,5,4),R(4,3,5),S(4,3,4)

Find, correct to the nearest degree, the angle between the planes. x+y−z=1,8x−11y+16z=5x + y - z = 1,8 x - 11 y + 16 z = 5x+y−z=1,8x−11y+16z=5

Identify the planes that are perpendicular. Select the correct answer.

Find the center and radius of the sphere. Select the correct answer. x2−4x−24+y2+16y+z2−12z=0x ^ { 2 } - 4 x - 24 + y ^ { 2 } + 16 y + z ^ { 2 } - 12 z = 0x2−4x−24+y2+16y+z2−12z=0

Classify the surface. 23x2+y2−z2−12y+12z=023 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 12 y + 12 z = 023x2+y2−z2−12y+12z=0

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes x=1,y=8x = 1 , y = 8x=1,y=8 , and z=4z = 4z=4 .

Find parametric equations for the line through the point (5,6,3)( 5,6,3 )(5,6,3) that is parallel to the plane x+y+z=−15x + y + z = - 15x+y+z=−15 and perpendicular to the line x=15+t,y=12−t,z=3tx = 15 + t , y = 12 - t , z = 3 tx=15+t,y=12−t,z=3t . Select the correct answer.

Sketch the plane in a three-dimensional space represented by the equation. z=−1z = - 1z=−1

Find parametric equations for the line through the point (3,4,5)( 3,4,5 )(3,4,5) that is parallel to the plane x+y+z=−15x + y + z = - 15x+y+z=−15 and perpendicular to the line x=15+t,y=12−t,z=3tx = 15 + t , y = 12 - t , z = 3 tx=15+t,y=12−t,z=3t .

Find parametric equations for the line through (−4,1,1) and (4,5,6). \text { Find parametric equations for the line through } ( - 4,1,1 ) \text { and } ( 4,5,6 ) \text {. } Find parametric equations for the line through (−4,1,1) and (4,5,6).

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xzx zxz -plane and 11 units to the right of it.

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xzx zxz -plane and 11 units to the right of it.

Find the distance (correct to two decimal places) between the given parallel planes. 6x+7y−2z=12 and 12x+14y−2z=706 x + 7 y - 2 z = 12 \text { and } 12 x + 14 y - 2 z = 706x+7y−2z=12 and 12x+14y−2z=70

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

Filters

Find that the midpoint of the line segment from $P 1 ( - 20,20,3 )$ to $P 2 ( 6,24,7 )$ .

Find the point at which the line given by the parametric equations below intersects the plane. $2 x + 4 y - 3 z = - 48 ; x = 10 + 7 t , \quad y = - 10 , z = 7 t$ Select the correct answer.

Find the volume of the parallelepiped with adjacent edges $P Q , P R$ , and $P S$ . $P ( 2,3,4 ) , Q ( 4,5,4 ) , R ( 4,3,5 ) , S ( 4,3,4 )$

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,8 x - 11 y + 16 z = 5$

Find the center and radius of the sphere. Select the correct answer. $x ^ { 2 } - 4 x - 24 + y ^ { 2 } + 16 y + z ^ { 2 } - 12 z = 0$

Classify the surface. $23 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 12 y + 12 z = 0$

Write inequalities to describe the solid rectangular box in the first octant bounded by the planes $x = 1 , y = 8$ , and $z = 4$ .

Find parametric equations for the line through the point $( 5,6,3 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$ . Select the correct answer.

Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$

Find parametric equations for the line through the point $( 3,4,5 )$ that is parallel to the plane $x + y + z = - 15$ and perpendicular to the line $x = 15 + t , y = 12 - t , z = 3 t$ .

$\text { Find parametric equations for the line through } ( - 4,1,1 ) \text { and } ( 4,5,6 ) \text {. }$

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$ -plane and 11 units to the right of it.

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$ -plane and 11 units to the right of it.

Find the distance (correct to two decimal places) between the given parallel planes. $6 x + 7 y - 2 z = 12 \text { and } 12 x + 14 y - 2 z = 70$