Question 1

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 }$ + $y ^ { 2 }$ + $z ^ { 2 }$ - 6x + 4y = 0

Accepted Answer

A)  ( -3, 2, 0), $\sqrt { 13 }$ 
B)  ( 3, -2, 0), 13 
C)  ( -3, 2, 0), 13 
D)  ( 3, -2, 0), $\sqrt { 13 }$ 
A)  ( -3, 2, 0), $\sqrt { 13 }$ 
B)  ( 3, -2, 0), 13 
C)  ( -3, 2, 0), 13 
D)  ( 3, -2, 0), $\sqrt { 13 }$

Question 2

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ .

Accepted Answer

A)  $z ^ { 2 } - x ^ { 2 } = - 8 y$ 
B)  $x ^ { 2 } + y ^ { 2 } = 8 z$ 
C)  $x ^ { 2 } + 8 y ^ { 2 } + z ^ { 2 } = 1$ 
D)  $z ^ { 2 } - 8 y ^ { 2 } = x$ 
E)  $z ^ { 2 } + x ^ { 2 } = - 8 y$ 
A)  $z ^ { 2 } - x ^ { 2 } = - 8 y$ 
B)  $x ^ { 2 } + y ^ { 2 } = 8 z$ 
C)  $x ^ { 2 } + 8 y ^ { 2 } + z ^ { 2 } = 1$ 
D)  $z ^ { 2 } - 8 y ^ { 2 } = x$ 
E)  $z ^ { 2 } + x ^ { 2 } = - 8 y$

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 N $60$ ${}^ { \circ }$ W at a speed of $80$ km/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 N $45$ ${}^ { \circ }$ E at an airspeed (speed in still air) of $250$ km/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 distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$ Round your answer to two decimal place.

Accepted Answer

A)  $4.79$ 
B)  $6.79$ 
C)  $4.29$ 
D)  $7.79$ 
E)  $5.79$ 
A)  $4.79$ 
B)  $6.79$ 
C)  $4.29$ 
D)  $7.79$ 
E)  $5.79$

Question 5

Reduce the equation to one of the standard forms. $z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 6 } - \frac { z ^ { 2 } } { 24 } = 1$ 
C)  $x ^ { 2 } + \frac { y ^ { 2 } } { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
D)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } - z ^ { 2 } = 1$ 
E)  $( x - 4 ) ^ { 2 } + \frac { y ^ { 2 } } { \left( \frac { 1 } { 4 } \right) ^ { 2 } } + z ^ { 2 } = 1$ 
A)  $x ^ { 2 } + y ^ { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 6 } - \frac { z ^ { 2 } } { 24 } = 1$ 
C)  $x ^ { 2 } + \frac { y ^ { 2 } } { 2 } - \frac { z ^ { 2 } } { 4 } = 1$ 
D)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } - z ^ { 2 } = 1$ 
E)  $( x - 4 ) ^ { 2 } + \frac { y ^ { 2 } } { \left( \frac { 1 } { 4 } \right) ^ { 2 } } + z ^ { 2 } = 1$

Question 6

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

Accepted Answer

A)  $\frac { \pm ( 2 \mathbf { i } - 8 \mathbf { j } ) } { 65 }$ 
B)  $\frac { \pm ( 2 \mathbf { i } + 8 \mathbf { j } ) } { 65 }$ 
C)  $\frac { \pm ( i + 8 \mathbf { j } ) } { \sqrt { 65 } }$ 
D)  $\frac { \pm ( \mathbf { i } + \mathbf { j } ) } { \sqrt { 65 } }$ 
E)  $\frac { \pm ( \mathbf { i } - 8 \mathbf { j } ) } { 65 }$ 
A)  $\frac { \pm ( 2 \mathbf { i } - 8 \mathbf { j } ) } { 65 }$ 
B)  $\frac { \pm ( 2 \mathbf { i } + 8 \mathbf { j } ) } { 65 }$ 
C)  $\frac { \pm ( i + 8 \mathbf { j } ) } { \sqrt { 65 } }$ 
D)  $\frac { \pm ( \mathbf { i } + \mathbf { j } ) } { \sqrt { 65 } }$ 
E)  $\frac { \pm ( \mathbf { i } - 8 \mathbf { j } ) } { 65 }$

Question 7

Find the distance from $( - 10,2 , - 8 )$ to the xy-planes.

Accepted Answer

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

Question 8

Calculate the given quantities if $\mathbf { a } = \mathbf { i } - 9 \mathbf { j } \text { and } \mathbf { b } = - 9 \mathbf { j } + 4 \mathbf { k }$ I. $\quad \mathbf { a } \cdot \mathbf { b }$
II $\mathbf { a } \times \mathbf { b }$
III. The angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree)

Accepted Answer

None

Question 9

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

Accepted Answer

A)  A hyperboloid of one sheet with center $( - 8,0,6 )$ and axis parallel to the z-axis. 
B)  A cone with axis parallel to the z-axis and vertex $( 0,6,6 )$ . 
C)  A circular paraboloid with vertex $( 0,8,6 )$ and axis the z-axis. 
A)  A hyperboloid of one sheet with center $( - 8,0,6 )$ and axis parallel to the z-axis. 
B)  A cone with axis parallel to the z-axis and vertex $( 0,6,6 )$ . 
C)  A circular paraboloid with vertex $( 0,8,6 )$ and axis the z-axis.

Question 10

Find an equation of the plane through the origin and parallel to the plane $- 3 x + 6 y - 3 z = - 1$

Accepted Answer

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

Question 11

Find parametric equations for the line through $( - 4,3,3 )$ and $( 4,5,6 )$

Accepted Answer

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

Question 12

Find the values of x such that the vectors $\{ 2 x , x , 3 \}$ and $\{ 6 , x , 9 \}$ are orthogonal.

Accepted Answer

A)  $x = 9 , x = - 9$ 
B)  $x = 3 , x = 9$ 
C)  $x = 3 , x = - 3$ 
D)  $x = - 3 , x = - 9$ 
E)  $x = - 3 , x = 9$ 
A)  $x = 9 , x = - 9$ 
B)  $x = 3 , x = 9$ 
C)  $x = 3 , x = - 3$ 
D)  $x = - 3 , x = - 9$ 
E)  $x = - 3 , x = 9$

Question 13

Find an equation of the sphere with center $( 6 , - 3,6 )$ that touches the xy-plane.

Accepted Answer

A)  $( x + 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$ 
B)  $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$ 
C)  $( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z + 6 ) ^ { 2 } = 36$ 
D)  $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 6$ 
E)  $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 40$ 
A)  $( x + 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$ 
B)  $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 36$ 
C)  $( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z + 6 ) ^ { 2 } = 36$ 
D)  $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 6$ 
E)  $( x - 6 ) ^ { 2 } + ( y + 3 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 40$

Question 14

Find an equation of the set of all points equidistant from the points $A ( 3 , - 4 , - 5 )$ and $B ( - 4,2 , - 10 )$

Accepted Answer

A)  $$7 x - 6 y + 5 z = - 35$$ 
B)  $7 x - 6 y + 5 z = - 35$ 
C)  $7 x - 6 y + 5 z = 35$ 
D)  $7 x - 6 y = 0$ 
E)  $- 7 x - 6 y + 5 z = - 35$ 
A)  $$7 x - 6 y + 5 z = - 35$$ 
B)  $7 x - 6 y + 5 z = - 35$ 
C)  $7 x - 6 y + 5 z = 35$ 
D)  $7 x - 6 y = 0$ 
E)  $- 7 x - 6 y + 5 z = - 35$

Question 15

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.

Accepted Answer

The answer of A woman walks due west on the...

Question 16

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \langle 7,5,1 \rangle , \mathbf { b } = ( - 5,2 , - 2 )$

Accepted Answer

A)  $( - 8 , - 12 , - 19 \}$ 
B)  $\{ - 12,39,9 \}$ 
C)  $\{ - 12,9,39 \}$ 
D)  $\{ 39 , - 12,9 \}$ 
E)  $( - 7 , - 23,25 )$ 
A)  $( - 8 , - 12 , - 19 \}$ 
B)  $\{ - 12,39,9 \}$ 
C)  $\{ - 12,9,39 \}$ 
D)  $\{ 39 , - 12,9 \}$ 
E)  $( - 7 , - 23,25 )$

Question 17

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,5 x - 13 y + 10 z = 5$

Accepted Answer

A)  $127 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $123 ^ { \circ }$ 
D)  $7 ^ { \circ }$ 
E)  none of these 
A)  $127 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $123 ^ { \circ }$ 
D)  $7 ^ { \circ }$ 
E)  none of these

Question 18

An ellipsoid is created by rotating the ellipse $4 x ^ { 2 } + y ^ { 2 } = 16$ about the x-axis. Find the equation of the ellipsoid.

Accepted Answer

A)  $16 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ 
B)  $4 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 16$ 
C)  $4 x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 16$ 
D)  $4 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ 
E)  $$4 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$$ 
A)  $16 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ 
B)  $4 x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 16$ 
C)  $4 x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 16$ 
D)  $4 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ 
E)  $$4 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$$

Question 19

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

Accepted Answer

A)  $y ^ { 2 } + 3 x ^ { 2 } = z$ 
B)  $z ^ { 2 } + 3 x ^ { 2 } = y$ 
C)  $z ^ { 2 } + 3 x ^ { 2 } + y ^ { 2 } = 1$ 
D)  $z ^ { 2 } - 3 x ^ { 2 } - y ^ { 2 } = 1$ 
E)  $z ^ { 2 } - 3 x ^ { 2 } = y$ 
A)  $y ^ { 2 } + 3 x ^ { 2 } = z$ 
B)  $z ^ { 2 } + 3 x ^ { 2 } = y$ 
C)  $z ^ { 2 } + 3 x ^ { 2 } + y ^ { 2 } = 1$ 
D)  $z ^ { 2 } - 3 x ^ { 2 } - y ^ { 2 } = 1$ 
E)  $z ^ { 2 } - 3 x ^ { 2 } = y$

Question 20

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 36 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

Accepted Answer

A)  $C ( - 1 , - 12 , - 13 ) , r = 6$ 
B)  $C ( - 1 , - 12 , - 13 ) , r = 6 \sqrt { 2 }$ 
C)  $C ( - 1 , - 12 , - 13 ) , r = \sqrt { 2 }$ 
D)  $C ( 1 , - 12 , - 13 ) , r = 6$ 
E)  none of these 
A)  $C ( - 1 , - 12 , - 13 ) , r = 6$ 
B)  $C ( - 1 , - 12 , - 13 ) , r = 6 \sqrt { 2 }$ 
C)  $C ( - 1 , - 12 , - 13 ) , r = \sqrt { 2 }$ 
D)  $C ( 1 , - 12 , - 13 ) , r = 6$ 
E)  none of these

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 }$ + $y ^ { 2 }$ + $z ^ { 2 }$ - 6x + 4y = 0

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ .

Find the distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$ Round your answer to two decimal place.

Reduce the equation to one of the standard forms. $z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24$

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

Find the distance from $( - 10,2 , - 8 )$ to the xy-planes.

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

Find an equation of the plane through the origin and parallel to the plane $- 3 x + 6 y - 3 z = - 1$

Find parametric equations for the line through $( - 4,3,3 )$ and $( 4,5,6 )$

Find the values of x such that the vectors $\{ 2 x , x , 3 \}$ and $\{ 6 , x , 9 \}$ are orthogonal.

Find an equation of the sphere with center $( 6 , - 3,6 )$ that touches the xy-plane.

Find an equation of the set of all points equidistant from the points $A ( 3 , - 4 , - 5 )$ and $B ( - 4,2 , - 10 )$

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.

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \langle 7,5,1 \rangle , \mathbf { b } = ( - 5,2 , - 2 )$

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,5 x - 13 y + 10 z = 5$

An ellipsoid is created by rotating the ellipse $4 x ^ { 2 } + y ^ { 2 } = 16$ about the x-axis. Find the equation of the ellipsoid.

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

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 36 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

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 center and the radius of the sphere that has the given equation. x2x ^ { 2 }x2 + y2y ^ { 2 }y2 + z2z ^ { 2 }z2 - 6x + 4y = 0

Find an equation for the surface consisting of all points that are equidistant from the point (0,−2,0)( 0 , - 2,0 )(0,−2,0) and the plane y=2y = 2y=2 .

Find the distance between the point (9,9,3)( 9,9,3 )(9,9,3) and the plane 15x−3y−8z=1015 x - 3 y - 8 z = 1015x−3y−8z=10 Round your answer to two decimal place.

Reduce the equation to one of the standard forms. z2=12x2+4y2−24z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24z2=12x2+4y2−24

Find the unit vectors that are parallel to the tangent line to the curve y=2x2y = 2 x ^ { 2 }y=2x2 at the point (2,8)( 2,8 )(2,8) .

Find the distance from (−10,2,−8)( - 10,2 , - 8 )(−10,2,−8) to the xy-planes.

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

Find an equation of the plane through the origin and parallel to the plane −3x+6y−3z=−1- 3 x + 6 y - 3 z = - 1−3x+6y−3z=−1

Find parametric equations for the line through (−4,3,3)( - 4,3,3 )(−4,3,3) and (4,5,6)( 4,5,6 )(4,5,6)

Find the values of x such that the vectors {2x,x,3}\{ 2 x , x , 3 \}{2x,x,3} and {6,x,9}\{ 6 , x , 9 \}{6,x,9} are orthogonal.

Find an equation of the sphere with center (6,−3,6)( 6 , - 3,6 )(6,−3,6) that touches the xy-plane.

Find an equation of the set of all points equidistant from the points A(3,−4,−5)A ( 3 , - 4 , - 5 )A(3,−4,−5) and B(−4,2,−10)B ( - 4,2 , - 10 )B(−4,2,−10)

A woman walks due west on the deck of a ship at 666 mi/h. The ship is moving north at a speed of 202020 mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Find the cross product a×b\mathbf { a } \times \mathbf { b }a×b . a=⟨7,5,1⟩,b=(−5,2,−2)\mathbf { a } = \langle 7,5,1 \rangle , \mathbf { b } = ( - 5,2 , - 2 )a=⟨7,5,1⟩,b=(−5,2,−2)

Find, correct to the nearest degree, the angle between the planes. x+y−z=1,5x−13y+10z=5x + y - z = 1,5 x - 13 y + 10 z = 5x+y−z=1,5x−13y+10z=5

An ellipsoid is created by rotating the ellipse 4x2+y2=164 x ^ { 2 } + y ^ { 2 } = 164x2+y2=16 about the x-axis. Find the equation of the ellipsoid.

Find an equation for the surface obtained by rotating the parabola y=3x2y = 3 x ^ { 2 }y=3x2 about the y-axis.

Find the center and radius of the sphere. x2−6x−36+y2+24y+z2−18z=0x ^ { 2 } - 6 x - 36 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0x2−6x−36+y2+24y+z2−18z=0

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

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 }$ + $y ^ { 2 }$ + $z ^ { 2 }$ - 6x + 4y = 0

Find an equation for the surface consisting of all points that are equidistant from the point $( 0 , - 2,0 )$ and the plane $y = 2$ .

Find the distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$ Round your answer to two decimal place.

Reduce the equation to one of the standard forms. $z ^ { 2 } = 12 x ^ { 2 } + 4 y ^ { 2 } - 24$

Find the unit vectors that are parallel to the tangent line to the curve $y = 2 x ^ { 2 }$ at the point $( 2,8 )$ .

Find the distance from $( - 10,2 , - 8 )$ to the xy-planes.

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

Find an equation of the plane through the origin and parallel to the plane $- 3 x + 6 y - 3 z = - 1$

Find parametric equations for the line through $( - 4,3,3 )$ and $( 4,5,6 )$

Find the values of x such that the vectors $\{ 2 x , x , 3 \}$ and $\{ 6 , x , 9 \}$ are orthogonal.

Find an equation of the sphere with center $( 6 , - 3,6 )$ that touches the xy-plane.

Find an equation of the set of all points equidistant from the points $A ( 3 , - 4 , - 5 )$ and $B ( - 4,2 , - 10 )$

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.

Find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \langle 7,5,1 \rangle , \mathbf { b } = ( - 5,2 , - 2 )$

Find, correct to the nearest degree, the angle between the planes. $x + y - z = 1,5 x - 13 y + 10 z = 5$

An ellipsoid is created by rotating the ellipse $4 x ^ { 2 } + y ^ { 2 } = 16$ about the x-axis. Find the equation of the ellipsoid.

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

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 36 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$