Question 1

Find a set of symmetric equations of the line that passes through the given points.  $( 5,0,5 ) , ( 1,6 , - 3 )$

Accepted Answer

A)  $\frac { x - 5 } { - 4 } = \frac { y } { 6 } = \frac { z - 5 } { - 8 }$ 
B)  $\frac { x - 5 } { - 4 } = \frac { y } { - 6 } = \frac { z - 5 } { - 8 }$ 
C)  $\frac { x - 5 } { - 4 } = \frac { y } { - 8 } = \frac { z - 5 } { 6 }$ 
D)  $\frac { x - 5 } { - 4 } = \frac { y } { 6 } = \frac { z + 5 } { - 8 }$ 
E)  $\frac { x + 5 } { - 4 } = \frac { y } { 6 } = \frac { z + 5 } { - 8 }$ 
A)  $\frac { x - 5 } { - 4 } = \frac { y } { 6 } = \frac { z - 5 } { - 8 }$ 
B)  $\frac { x - 5 } { - 4 } = \frac { y } { - 6 } = \frac { z - 5 } { - 8 }$ 
C)  $\frac { x - 5 } { - 4 } = \frac { y } { - 8 } = \frac { z - 5 } { 6 }$ 
D)  $\frac { x - 5 } { - 4 } = \frac { y } { 6 } = \frac { z + 5 } { - 8 }$ 
E)  $\frac { x + 5 } { - 4 } = \frac { y } { 6 } = \frac { z + 5 } { - 8 }$ A

Question 2

Find the dot product of u and v. u = $( - 1,9 , - 5 )$ ,v = $( 3,2 , - 6 )$

Accepted Answer

A) 45 
B)  $( 11 , - 11,2 )$ 
C)  $( 2,18 , - 11 )$ 
D) -15 
E) 2 
A) 45 
B)  $( 11 , - 11,2 )$ 
C)  $( 2,18 , - 11 )$ 
D) -15 
E) 2 A

Question 3

Find the standard form of the equation of the sphere with the given characteristics. Center: (-6,-6,6);radius 2

Accepted Answer

A)  $( x - 6 ) ^ { 2 } - ( y - 6 ) ^ { 2 } - ( z + 6 ) ^ { 2 } = 4$ 
B)  $( x - 6 ) ^ { 2 } + ( y - 6 ) ^ { 2 } + ( z + 6 ) ^ { 2 } = 4$ 
C)  $( x + 6 ) ^ { 2 } + ( y + 6 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 4$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ 
E)  $( x + 6 ) ^ { 2 } - ( y + 6 ) ^ { 2 } - ( z - 6 ) ^ { 2 } = 4$ 
A)  $( x - 6 ) ^ { 2 } - ( y - 6 ) ^ { 2 } - ( z + 6 ) ^ { 2 } = 4$ 
B)  $( x - 6 ) ^ { 2 } + ( y - 6 ) ^ { 2 } + ( z + 6 ) ^ { 2 } = 4$ 
C)  $( x + 6 ) ^ { 2 } + ( y + 6 ) ^ { 2 } + ( z - 6 ) ^ { 2 } = 4$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ 
E)  $( x + 6 ) ^ { 2 } - ( y + 6 ) ^ { 2 } - ( z - 6 ) ^ { 2 } = 4$ C

Question 4

Find u × v and show that it is orthogonal to both u and v.  $\begin{array} { l } 
\mathbf { u } = 8 \mathbf { k } \
\mathbf { v } = - \mathbf { i } + 4 \mathbf { j } + \mathbf { k }
\end{array}$

Accepted Answer

A)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = 4 \mathbf { i } - 8 \mathbf { j } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
B)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = - 32 \mathbf { i } - 8 \mathbf { k } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
C)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = - 32 \mathbf { i } - 8 \mathbf { j } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
D)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = - 32 \mathbf { j } - 8 \mathbf { k } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
E)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = 8 \mathbf { i } - 8 \mathbf { j } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
A)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = 4 \mathbf { i } - 8 \mathbf { j } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
B)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = - 32 \mathbf { i } - 8 \mathbf { k } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
C)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = - 32 \mathbf { i } - 8 \mathbf { j } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
D)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = - 32 \mathbf { j } - 8 \mathbf { k } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
E)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = 8 \mathbf { i } - 8 \mathbf { j } \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$

Question 5

Determine whether the planes are parallel,orthogonal,or neither. 
5x - 4y + z = -2
3x + 4y + z = -5

Accepted Answer

A) parallel 
B) orthogonal 
C) neither 
A) parallel 
B) orthogonal 
C) neither

Question 6

Find the triple scalar product.  $\mathbf { u } = 10 \mathbf { i } + 11 \mathbf { j } + 11 \mathbf { k } , \mathbf { v } = 9 \mathbf { i } + 10 \mathbf { j } , \mathbf { w } = 9 \mathbf { i } + 11 \mathbf { j } + 20 \mathbf { k }$

Accepted Answer

A)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 120$ 
B)  $\mathbf { u } ( \mathbf { v } \times \mathbf { w } ) = 121$ 
C)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 119$ 
D)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 118$ 
E)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 117$ 
A)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 120$ 
B)  $\mathbf { u } ( \mathbf { v } \times \mathbf { w } ) = 121$ 
C)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 119$ 
D)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 118$ 
E)  $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = 117$

Question 7

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied.  $y z > 5$

Accepted Answer

A) Octants I,II,VII,or VIII 
B) Octants I,II,III,IV 
C) Octant II 
D) Octants II,VI,VIII 
E) Octants I,II,III 
A) Octants I,II,VII,or VIII 
B) Octants I,II,III,IV 
C) Octant II 
D) Octants II,VI,VIII 
E) Octants I,II,III

Question 8

Find a set of parametric equations of the line that passes through the given points.   $( - 4,7,13 ) , ( 1 , - 4,17 )$

Accepted Answer

A)  $x = - 4 + 5 t , y = 7 - 11 t , z = 13 - 4 t$ 
B)  $x = - 4 - 5 t , y = 7 - 11 t , z = 13 - 4 t$ 
C)  $x = - 4 + 5 t , y = - 11 t , z = 13 + 4 t$ 
D)  $x = - 4 + 5 t , y = 7 - 11 t , z = 4 t$ 
E)  $x = - 4 + 5 t , y = 7 - 11 t , z = 13 + 4 t$ 
A)  $x = - 4 + 5 t , y = 7 - 11 t , z = 13 - 4 t$ 
B)  $x = - 4 - 5 t , y = 7 - 11 t , z = 13 - 4 t$ 
C)  $x = - 4 + 5 t , y = - 11 t , z = 13 + 4 t$ 
D)  $x = - 4 + 5 t , y = 7 - 11 t , z = 4 t$ 
E)  $x = - 4 + 5 t , y = 7 - 11 t , z = 13 + 4 t$

Question 9

Find the coordinates of the point. 
The point is located in the yz-plane,four unit to the right of the xz-plane,and six units above the xy-plane.

Accepted Answer

A)  $x = 6 , y = 4 , z = 6$ 
B)  $x = 4 , y = 4 , z = 6$ 
C)  $x = 4 , y = 0 , z = 6$ 
D)  $x = 0 , y = 4 , z = 0$ 
E)  $x = 0 , y = 4 , z = 6$ 
A)  $x = 6 , y = 4 , z = 6$ 
B)  $x = 4 , y = 4 , z = 6$ 
C)  $x = 4 , y = 0 , z = 6$ 
D)  $x = 0 , y = 4 , z = 0$ 
E)  $x = 0 , y = 4 , z = 6$

Question 10

Write the component form of the vector described below. Initial point: (3,0,1)
Terminal point: (-7,-6,-5)

Accepted Answer

A)  $( - 10 , - 6 , - 6 )$ 
B)  $( 10 , - 6 , - 6 )$ 
C)  $( - 10 , - 6,6 )$ 
D)  $( - 10,6 , - 6 )$ 
E)  $( 10,6,6 )$ 
A)  $( - 10 , - 6 , - 6 )$ 
B)  $( 10 , - 6 , - 6 )$ 
C)  $( - 10 , - 6,6 )$ 
D)  $( - 10,6 , - 6 )$ 
E)  $( 10,6,6 )$

Question 11

Find a unit vector in the opposite direction of u. u = $( - 8 , - 11,1 )$

Accepted Answer

A)  $\frac { 1 } { \sqrt { 20 } } ( - 8 , - 11,1 )$ 
B)  $\frac { 1 } { \sqrt { 186 } } ( 8,11 , - 1 )$ 
C)  $( - 8 , - 11,1 )$ 
D)  $\sqrt { 186 } ( 8,11 , - 1 )$ 
E)  $\sqrt { 20 } ( - 8 , - 11,1 )$ 
A)  $\frac { 1 } { \sqrt { 20 } } ( - 8 , - 11,1 )$ 
B)  $\frac { 1 } { \sqrt { 186 } } ( 8,11 , - 1 )$ 
C)  $( - 8 , - 11,1 )$ 
D)  $\sqrt { 186 } ( 8,11 , - 1 )$ 
E)  $\sqrt { 20 } ( - 8 , - 11,1 )$

Question 12

Find the volume of the parallelepiped with the given vertices 
 $A ( 0,0,0 ) , B ( 2,0,0 ) , C ( 0,3,0 ) , D ( 2,3,0 ) , E ( 0,0,5 ) , F ( 2,0,5 ) , G ( 0,3,5 ) , H ( 2,3,5 )$ .

Accepted Answer

A) 15 
B) 30 
C) 7.5 
D) 60 
E) 120 
A) 15 
B) 30 
C) 7.5 
D) 60 
E) 120

Question 13

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 6 x - 12 y - 6 z + 29 = 0$

Accepted Answer

A) center: (3,-6,-3);radius: 25. 
B) center: (-3,6,3);radius: 5 
C) center: (3,-6,-3);radius: 25 
D) center: (-3,-6,-3);radius: 5 
E) center: (3,-6,-3);radius: 5 
A) center: (3,-6,-3);radius: 25. 
B) center: (-3,6,3);radius: 5 
C) center: (3,-6,-3);radius: 25 
D) center: (-3,-6,-3);radius: 5 
E) center: (3,-6,-3);radius: 5

Question 14

Find the area of the parallelogram that has the vectors as adjacent sides.  $\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )$

Accepted Answer

A) 15 
B)  $0$ 
C)  $\sqrt { 3 }$ 
D) 21 
E)  $5 \sqrt { 35 }$ 
A) 15 
B)  $0$ 
C)  $\sqrt { 3 }$ 
D) 21 
E)  $5 \sqrt { 35 }$

Question 15

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = 5i+5j+3k,v = 8i+8j+2k

Accepted Answer

A) 44.3° 
B) 77° 
C) 45.7° 
D) 13° 
E) 90° 
A) 44.3° 
B) 77° 
C) 45.7° 
D) 13° 
E) 90°

Question 16

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. ) 
Point: $( - 6,2,0 )$ Parallel to: $\mathrm { v } = \frac { 3 } { 7 } \mathrm { i } + \frac { - 3 } { 7 } \mathrm { j } - 5 \mathrm { k }$

Accepted Answer

A) Parametric equations: $x = - 6 - 3 t , y = 4 - 3 t , z = - 5 t$ 
B) Parametric equations: $x = - 2 + 3 t , y = 5 - 3 t , z = - 5 t$ 
C) Parametric equations: $x = - 5 t , y = 5 - 3 t , z = - 5 t$ 
D) Parametric equations: $x = - 6 + 3 t , y = 2 - 3 t , z = - 35 t$ 
E) Parametric equations: $x = 3 t , y = - 5 t , z = - 3 t$ 
A) Parametric equations: $x = - 6 - 3 t , y = 4 - 3 t , z = - 5 t$ 
B) Parametric equations: $x = - 2 + 3 t , y = 5 - 3 t , z = - 5 t$ 
C) Parametric equations: $x = - 5 t , y = 5 - 3 t , z = - 5 t$ 
D) Parametric equations: $x = - 6 + 3 t , y = 2 - 3 t , z = - 35 t$ 
E) Parametric equations: $x = 3 t , y = - 5 t , z = - 3 t$

Question 17

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. ) 
Point: $( - 7,2,2 )$ Parallel to: $\mathbf { v } = 3 \mathbf { i } + 6 \mathbf { j } - 2 \mathbf { k }$

Accepted Answer

A) Parametric equations: $x = 3 + - 7 t , y = 2 + 6 t , z = 2 - 2 t$ 
B) Parametric equations: $x = - 7 + 3 t , y = 6 + 2 t , z = 2 - 2 t$ 
C) Parametric equations: $x = - 7 + 2 t , y = 2 + 6 t , z = 2 - 2 t$ 
D) Parametric equations: $x = - 7 + 3 t , y = 2 + 6 t , z = 2 - 2 t$ 
E) Parametric equations: $x = 2 + 3 t , y = 2 + 6 t , z = 2 - 2 t$ 
A) Parametric equations: $x = 3 + - 7 t , y = 2 + 6 t , z = 2 - 2 t$ 
B) Parametric equations: $x = - 7 + 3 t , y = 6 + 2 t , z = 2 - 2 t$ 
C) Parametric equations: $x = - 7 + 2 t , y = 2 + 6 t , z = 2 - 2 t$ 
D) Parametric equations: $x = - 7 + 3 t , y = 2 + 6 t , z = 2 - 2 t$ 
E) Parametric equations: $x = 2 + 3 t , y = 2 + 6 t , z = 2 - 2 t$

Question 18

Find the distance between the points. (0,0,0), (-9,6,8)

Accepted Answer

A) 5 
B) 23 
C)  $\sqrt { 5 }$ 
D)  $2 \sqrt { 181 }$ 
E)  $\sqrt { 181 }$ 
A) 5 
B) 23 
C)  $\sqrt { 5 }$ 
D)  $2 \sqrt { 181 }$ 
E)  $\sqrt { 181 }$

Question 19

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

Accepted Answer

A) Center: $( 2,0 , - 2 )$ Radius: $2$ 
B) Center: $( 2,2,2 )$ Radius: $2$ 
C) Center: $( 2,0,2 )$ Radius: $2$ 
D) Center: $( 2,0,0 )$ Radius: $2$ 
E) Center: $( 2 , - 2,0 )$ Radius: $2$ 
A) Center: $( 2,0 , - 2 )$ Radius: $2$ 
B) Center: $( 2,2,2 )$ Radius: $2$ 
C) Center: $( 2,0,2 )$ Radius: $2$ 
D) Center: $( 2,0,0 )$ Radius: $2$ 
E) Center: $( 2 , - 2,0 )$ Radius: $2$

Question 20

Determine whether the triangle is a right triangle,an isosceles triangle,or neither.  $( 7,3,2 ) , ( 9,1,1 ) , ( 5,5,1 )$

Accepted Answer

A) Isosceles triangle 
B) Right triangle 
C) Neither Isosceles nor Right 
A) Isosceles triangle 
B) Right triangle 
C) Neither Isosceles nor Right

Find a set of symmetric equations of the line that passes through the given points. $( 5,0,5 ) , ( 1,6 , - 3 )$

Find the dot product of u and v. u = $( - 1,9 , - 5 )$ ,v = $( 3,2 , - 6 )$

Find the standard form of the equation of the sphere with the given characteristics. Center: (-6,-6,6);radius 2

Find u × v and show that it is orthogonal to both u and v. =8 =-+4+

Determine whether the planes are parallel,orthogonal,or neither. 5x - 4y + z = -2 3x + 4y + z = -5

Find the triple scalar product. $\mathbf { u } = 10 \mathbf { i } + 11 \mathbf { j } + 11 \mathbf { k } , \mathbf { v } = 9 \mathbf { i } + 10 \mathbf { j } , \mathbf { w } = 9 \mathbf { i } + 11 \mathbf { j } + 20 \mathbf { k }$

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $y z > 5$

Find a set of parametric equations of the line that passes through the given points. $( - 4,7,13 ) , ( 1 , - 4,17 )$

Find the coordinates of the point. The point is located in the yz-plane,four unit to the right of the xz-plane,and six units above the xy-plane.

Write the component form of the vector described below. Initial point: (3,0,1) Terminal point: (-7,-6,-5)

Find a unit vector in the opposite direction of u. u = $( - 8 , - 11,1 )$

Find the volume of the parallelepiped with the given vertices $A ( 0,0,0 ) , B ( 2,0,0 ) , C ( 0,3,0 ) , D ( 2,3,0 ) , E ( 0,0,5 ) , F ( 2,0,5 ) , G ( 0,3,5 ) , H ( 2,3,5 )$ .

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 6 x - 12 y - 6 z + 29 = 0$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )$

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = 5i+5j+3k,v = 8i+8j+2k

Find a set of parametric equations for the line through the point and parallel to the specified vector or line.(For each line,write the direction numbers as integers. ) Point: $( - 7,2,2 )$ Parallel to: $\mathbf { v } = 3 \mathbf { i } + 6 \mathbf { j } - 2 \mathbf { k }$

Find the distance between the points. (0,0,0), (-9,6,8)

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

Determine whether the triangle is a right triangle,an isosceles triangle,or neither. $( 7,3,2 ) , ( 9,1,1 ) , ( 5,5,1 )$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Limits and An Introduction To Calculus

Filters

Exam 11: Analytic Geometry In Three Dimensions

Find a set of symmetric equations of the line that passes through the given points. (5,0,5),(1,6,−3)( 5,0,5 ) , ( 1,6 , - 3 )(5,0,5),(1,6,−3)

Find the dot product of u and v. u = (−1,9,−5)( - 1,9 , - 5 )(−1,9,−5) ,v = (3,2,−6)( 3,2 , - 6 )(3,2,−6)

Find the standard form of the equation of the sphere with the given characteristics. Center: (-6,-6,6);radius 2

Find u × v and show that it is orthogonal to both u and v. =8 =-+4+

Determine whether the planes are parallel,orthogonal,or neither. 5x - 4y + z = -2 3x + 4y + z = -5

Find the triple scalar product. u=10i+11j+11k,v=9i+10j,w=9i+11j+20k\mathbf { u } = 10 \mathbf { i } + 11 \mathbf { j } + 11 \mathbf { k } , \mathbf { v } = 9 \mathbf { i } + 10 \mathbf { j } , \mathbf { w } = 9 \mathbf { i } + 11 \mathbf { j } + 20 \mathbf { k }u=10i+11j+11k,v=9i+10j,w=9i+11j+20k

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. yz>5y z > 5yz>5

Find a set of parametric equations of the line that passes through the given points. (−4,7,13),(1,−4,17)( - 4,7,13 ) , ( 1 , - 4,17 )(−4,7,13),(1,−4,17)

Find the coordinates of the point. The point is located in the yz-plane,four unit to the right of the xz-plane,and six units above the xy-plane.

Write the component form of the vector described below. Initial point: (3,0,1) Terminal point: (-7,-6,-5)

Find a unit vector in the opposite direction of u. u = (−8,−11,1)( - 8 , - 11,1 )(−8,−11,1)

Find the center and radius of the sphere. x2+y2+z2+6x−12y−6z+29=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 6 x - 12 y - 6 z + 29 = 0x2+y2+z2+6x−12y−6z+29=0

Find the area of the parallelogram that has the vectors as adjacent sides. u=(−1,4,−3),v=(−3,3,4)\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )u=(−1,4,−3),v=(−3,3,4)

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = 5i+5j+3k,v = 8i+8j+2k

Find the distance between the points. (0,0,0), (-9,6,8)

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

Determine whether the triangle is a right triangle,an isosceles triangle,or neither. (7,3,2),(9,1,1),(5,5,1)( 7,3,2 ) , ( 9,1,1 ) , ( 5,5,1 )(7,3,2),(9,1,1),(5,5,1)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Limits and An Introduction To Calculus

Filters

Find a set of symmetric equations of the line that passes through the given points. $( 5,0,5 ) , ( 1,6 , - 3 )$

Find the dot product of u and v. u = $( - 1,9 , - 5 )$ ,v = $( 3,2 , - 6 )$

Find the triple scalar product. $\mathbf { u } = 10 \mathbf { i } + 11 \mathbf { j } + 11 \mathbf { k } , \mathbf { v } = 9 \mathbf { i } + 10 \mathbf { j } , \mathbf { w } = 9 \mathbf { i } + 11 \mathbf { j } + 20 \mathbf { k }$

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $y z > 5$

Find a set of parametric equations of the line that passes through the given points. $( - 4,7,13 ) , ( 1 , - 4,17 )$

Find a unit vector in the opposite direction of u. u = $( - 8 , - 11,1 )$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 6 x - 12 y - 6 z + 29 = 0$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( - 1,4 , - 3 ) , \mathbf { v } = ( - 3,3,4 )$

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

Determine whether the triangle is a right triangle,an isosceles triangle,or neither. $( 7,3,2 ) , ( 9,1,1 ) , ( 5,5,1 )$