Question 1

Find the lengths of the sides of the triangle with the indicated vertices. $( 4,0,3 ) , ( 4,2,2 ) , ( 1,0,4 )$

Accepted Answer

A) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 5 }$ 
B) d1= $\sqrt { 10 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$ 
C) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 5 }$ ,d3= $\sqrt { 17 }$ 
D) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$ 
E) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 17 }$ ,d3= $\sqrt { 17 }$ 
A) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 5 }$ 
B) d1= $\sqrt { 10 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$ 
C) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 5 }$ ,d3= $\sqrt { 17 }$ 
D) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$ 
E) d1= $\sqrt { 5 }$ ,d2= $\sqrt { 17 }$ ,d3= $\sqrt { 17 }$

Question 2

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 7 , - 9,3 )$ ,v = $( - 21 , - 27,9 )$

Accepted Answer

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

Question 3

Find symmetric equations for the line through the point and parallel to the specified vector.
(9,-6,-5),parallel to $( 7,2 , - 7 )$

Accepted Answer

Answers may vary.One

Question 4

Determine whether the planes are parallel,orthogonal,or neither. 
5x - 6y - 6z = 0
3x - 4y + 2z = -4

Accepted Answer

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

Question 5

Determine whether the triangle is a right triangle,an isosceles triangle,or neither.  $( 8,1,1 ) , ( 3 , - 6 , - 1 ) , ( 1 , - 1 , - 8 )$

Accepted Answer

A) Neither Isosceles nor Right 
B) Isosceles triangle 
C) Right triangle 
D) Right triangle and isosceles triangle 
A) Neither Isosceles nor Right 
B) Isosceles triangle 
C) Right triangle 
D) Right triangle and isosceles triangle

Question 6

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 5 , - 5,5 )$ ,v =  $( - 20 , - 20,20 )$

Accepted Answer

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

Question 7

Find the magnitude of the vector described below. Initial point: (3,-1,2)
Terminal point: (4,-7,4)

Accepted Answer

A) 3 
B)  $3$ 
C) 27 
D)  $\sqrt { 41 }$ 
E)  $6$ 
A) 3 
B)  $3$ 
C) 27 
D)  $\sqrt { 41 }$ 
E)  $6$

Question 8

Find the general form of the equation of the plane passing through the three points.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (3,1,6), (-6,-1,6), (3,5,-6)

Accepted Answer

A) 2x - 9y - 3z + 21 = 0 
B) 2x - 9y - 3z = 0 
C) 3x + y + 6z = 0 
D) 2x - 9y - 3z - 21 = 0 
E) 3x + y + 6z - 252 = 0 
A) 2x - 9y - 3z + 21 = 0 
B) 2x - 9y - 3z = 0 
C) 3x + y + 6z = 0 
D) 2x - 9y - 3z - 21 = 0 
E) 3x + y + 6z - 252 = 0

Question 9

The weight of a crate is 200 newtons.Find the tension in each of the supporting cables shown in the figure.The coordinates of the points A,B,C,and D are given below the figure.Round to the nearest newton.   [Figure not necessarily to scale.] point A = (0,0,-80),point B = (170,0,0),point C = (-30,40,0),point D = (0,-120,0) $\vec { T } + m \vec { g } = 0$

Accepted Answer

A) cable AB = 55;cable AC = 156;cable AD = 80 
B) cable AB = 156;cable AC = 55;cable AD = 80 
C) cable AB = 55;cable AC = 80;cable AD = 156 
D) cable AB = 80;cable AC = 156;cable AD = 55 
E) cable AB = 156;cable AC = 80;cable AD = 55 
A) cable AB = 55;cable AC = 156;cable AD = 80 
B) cable AB = 156;cable AC = 55;cable AD = 80 
C) cable AB = 55;cable AC = 80;cable AD = 156 
D) cable AB = 80;cable AC = 156;cable AD = 55 
E) cable AB = 156;cable AC = 80;cable AD = 55

Question 10

Find a unit vector in the direction of the vector described below. Initial point: (4,-6,-1)
Terminal point: (2,5,-1)

Accepted Answer

A)  $( - 2,11,0 )$ 
B)  $\sqrt { 13 } ( - 2,11,0 )$ 
C)  $\sqrt { 125 } ( - 2,11,0 )$ 
D)  $\frac { 1 } { \sqrt { 125 } } ( - 2,11,0 )$ 
E)  $\frac { 1 } { \sqrt { 13 } } ( - 2,11,0 )$ 
A)  $( - 2,11,0 )$ 
B)  $\sqrt { 13 } ( - 2,11,0 )$ 
C)  $\sqrt { 125 } ( - 2,11,0 )$ 
D)  $\frac { 1 } { \sqrt { 125 } } ( - 2,11,0 )$ 
E)  $\frac { 1 } { \sqrt { 13 } } ( - 2,11,0 )$

Question 11

Determine the values of c such that $\| c u \| = 9$ ,where $\mathbf { u } = \mathbf { i } + 8 \mathbf { j } + 9 \mathbf { k }$ .

Accepted Answer

A)  $\pm 10 \sqrt { 146 }$ 
B)  $\frac { 9 } { 146 }$ 
C)  $\pm \frac { 10 \sqrt { 146 } } { 146 }$ 
D)  $- \frac { 9 } { 146 }$ 
E)  $\pm \frac { 9 \sqrt { 146 } } { 146 }$ 
A)  $\pm 10 \sqrt { 146 }$ 
B)  $\frac { 9 } { 146 }$ 
C)  $\pm \frac { 10 \sqrt { 146 } } { 146 }$ 
D)  $- \frac { 9 } { 146 }$ 
E)  $\pm \frac { 9 \sqrt { 146 } } { 146 }$

Question 12

Find the triple scalar product u · (v × w) for the vectors  $\mathbf { u } = ( 7,9,4 ) , \mathbf { v } = ( 9 , - 4,1 ) , \mathbf { w } = ( 8,8 , - 4 )$

Accepted Answer

A) 768 
B) 868 
C) 993 
D) 618 
E) 1193 
A) 768 
B) 868 
C) 993 
D) 618 
E) 1193

Question 13

Use the vectors u and v to find u × (-v). $\mathbf { u } = 7 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } \quad \mathbf { v } = 6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$

Accepted Answer

A)  $\mathbf { u } \times ( - \mathbf { v } ) = 47 \mathbf { i } - 55 \mathbf { j } - 48 \mathbf { k }$ 
B)  $\mathbf { u } \times ( - \mathbf { v } ) = 55 \mathbf { i } - 55 \mathbf { j } - 48 \mathbf { k }$ 
C)  $\mathbf { u } \times ( - \mathbf { v } ) = 47 \mathbf { i } + 47 \mathbf { j } - 48 \mathbf { k }$ 
D)  $\mathbf { u } \times ( - \mathbf { v } ) = 55 \mathbf { i } + 47 \mathbf { j } + 48 \mathbf { k }$ 
E)  $\mathbf { u } \times ( - \mathbf { v } ) = 48 \mathbf { i } - 55 \mathbf { j } + 47 \mathbf { k }$ 
A)  $\mathbf { u } \times ( - \mathbf { v } ) = 47 \mathbf { i } - 55 \mathbf { j } - 48 \mathbf { k }$ 
B)  $\mathbf { u } \times ( - \mathbf { v } ) = 55 \mathbf { i } - 55 \mathbf { j } - 48 \mathbf { k }$ 
C)  $\mathbf { u } \times ( - \mathbf { v } ) = 47 \mathbf { i } + 47 \mathbf { j } - 48 \mathbf { k }$ 
D)  $\mathbf { u } \times ( - \mathbf { v } ) = 55 \mathbf { i } + 47 \mathbf { j } + 48 \mathbf { k }$ 
E)  $\mathbf { u } \times ( - \mathbf { v } ) = 48 \mathbf { i } - 55 \mathbf { j } + 47 \mathbf { k }$

Question 14

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (5,5,-2)and (4,2,1)and is perpendicular to the plane -3x - 2y + 4z = 1.

Accepted Answer

A) 6x + 5y + 7z - 41 = 0 
B) 3x + 2y - 4z + 33 = 0 
C) 6x - 5y + 7z - 9 = 0 
D) 3x + 2y - 4z = 0 
E) 7x + 6y + 5z - 55 = 0 
A) 6x + 5y + 7z - 41 = 0 
B) 3x + 2y - 4z + 33 = 0 
C) 6x - 5y + 7z - 9 = 0 
D) 3x + 2y - 4z = 0 
E) 7x + 6y + 5z - 55 = 0

Question 15

Find the distance between the points. $( - 5,1 , - 4 ) , ( 8,0 , - 6 )$

Accepted Answer

A)  $4 \sqrt { 11 }$ units 
B)  $5 \sqrt { 7 }$ units 
C)  $\sqrt { 174 }$ units 
D)  $\sqrt { 178 }$ units 
E)  $\sqrt { 177 }$ units 
A)  $4 \sqrt { 11 }$ units 
B)  $5 \sqrt { 7 }$ units 
C)  $\sqrt { 174 }$ units 
D)  $\sqrt { 178 }$ units 
E)  $\sqrt { 177 }$ units

Question 16

Find the distance between the points. (-2,4,-1,), (-5,0,-5)

Accepted Answer

A) -9 
B) 11 
C)  $\sqrt { 11 }$ 
D)  $2 \sqrt { 41 }$ 
E)  $\sqrt { 41 }$ 
A) -9 
B) 11 
C)  $\sqrt { 11 }$ 
D)  $2 \sqrt { 41 }$ 
E)  $\sqrt { 41 }$

Question 17

Find u × v. $\mathbf { u } = ( - 5 , - 3,5 ) \quad \mathbf { v } = ( 4 , - 8,0 )$

Accepted Answer

A)  $( 40 , - 20,52 )$ 
B)  $4$ 
C)  $( 40,20,52 )$ 
D)  $- 44$ 
E)  $( - 20,24,0 )$ 
A)  $( 40 , - 20,52 )$ 
B)  $4$ 
C)  $( 40,20,52 )$ 
D)  $- 44$ 
E)  $( - 20,24,0 )$

Question 18

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 6 , - 6,5 )$ ,v =  $( 8,0,9 )$

Accepted Answer

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

Question 19

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 14 z = 0$

Accepted Answer

A) center: (0,0,0);radius: 49 
B) center: (0,0,7);radius: 49 
C) center: (0,0,-7);radius: 7 
D) center: (0,0,7);radius: 7 
E) center: (0,0,0);radius: 0 
A) center: (0,0,0);radius: 49 
B) center: (0,0,7);radius: 49 
C) center: (0,0,-7);radius: 7 
D) center: (0,0,7);radius: 7 
E) center: (0,0,0);radius: 0

Question 20

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. 
Point: $( 3,0,0 )$ Perpendicular to: $x = 3 - t , y = 2 - 6 t , z = 4 + t$

Accepted Answer

A)  $- x - 6 y - z + 3 = 0$ 
B)  $x - 6 y + z + 3 = 0$ 
C)  $- x + 6 y + z + 3 = 0$ 
D)  $- x - 6 y + z + 3 = 0$ 
E)  $- x - 6 y - z - 3 = 0$ 
A)  $- x - 6 y - z + 3 = 0$ 
B)  $x - 6 y + z + 3 = 0$ 
C)  $- x + 6 y + z + 3 = 0$ 
D)  $- x - 6 y + z + 3 = 0$ 
E)  $- x - 6 y - z - 3 = 0$

Find the lengths of the sides of the triangle with the indicated vertices. $( 4,0,3 ) , ( 4,2,2 ) , ( 1,0,4 )$

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 7 , - 9,3 )$ ,v = $( - 21 , - 27,9 )$

Find symmetric equations for the line through the point and parallel to the specified vector. (9,-6,-5),parallel to $( 7,2 , - 7 )$

Determine whether the planes are parallel,orthogonal,or neither. 5x - 6y - 6z = 0 3x - 4y + 2z = -4

Determine whether the triangle is a right triangle,an isosceles triangle,or neither. $( 8,1,1 ) , ( 3 , - 6 , - 1 ) , ( 1 , - 1 , - 8 )$

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 5 , - 5,5 )$ ,v = $( - 20 , - 20,20 )$

Find the magnitude of the vector described below. Initial point: (3,-1,2) Terminal point: (4,-7,4)

Find the general form of the equation of the plane passing through the three points.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (3,1,6), (-6,-1,6), (3,5,-6)

Find a unit vector in the direction of the vector described below. Initial point: (4,-6,-1) Terminal point: (2,5,-1)

Determine the values of c such that $\| c u \| = 9$ ,where $\mathbf { u } = \mathbf { i } + 8 \mathbf { j } + 9 \mathbf { k }$ .

Find the triple scalar product u · (v × w) for the vectors $\mathbf { u } = ( 7,9,4 ) , \mathbf { v } = ( 9 , - 4,1 ) , \mathbf { w } = ( 8,8 , - 4 )$

Use the vectors u and v to find u × (-v). $\mathbf { u } = 7 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } \quad \mathbf { v } = 6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (5,5,-2)and (4,2,1)and is perpendicular to the plane -3x - 2y + 4z = 1.

Find the distance between the points. $( - 5,1 , - 4 ) , ( 8,0 , - 6 )$

Find the distance between the points. (-2,4,-1,), (-5,0,-5)

Find u × v. $\mathbf { u } = ( - 5 , - 3,5 ) \quad \mathbf { v } = ( 4 , - 8,0 )$

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 6 , - 6,5 )$ ,v = $( 8,0,9 )$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 14 z = 0$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: $( 3,0,0 )$ Perpendicular to: $x = 3 - t , y = 2 - 6 t , z = 4 + t$

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 the lengths of the sides of the triangle with the indicated vertices. (4,0,3),(4,2,2),(1,0,4)( 4,0,3 ) , ( 4,2,2 ) , ( 1,0,4 )(4,0,3),(4,2,2),(1,0,4)

Determine whether u and v are parallel,orthogonal,or neither. u = (−7,−9,3)( - 7 , - 9,3 )(−7,−9,3) ,v = (−21,−27,9)( - 21 , - 27,9 )(−21,−27,9)

Find symmetric equations for the line through the point and parallel to the specified vector. (9,-6,-5),parallel to (7,2,−7)( 7,2 , - 7 )(7,2,−7)

Determine whether the planes are parallel,orthogonal,or neither. 5x - 6y - 6z = 0 3x - 4y + 2z = -4

Determine whether the triangle is a right triangle,an isosceles triangle,or neither. (8,1,1),(3,−6,−1),(1,−1,−8)( 8,1,1 ) , ( 3 , - 6 , - 1 ) , ( 1 , - 1 , - 8 )(8,1,1),(3,−6,−1),(1,−1,−8)

Determine whether u and v are parallel,orthogonal,or neither. u = (−5,−5,5)( - 5 , - 5,5 )(−5,−5,5) ,v = (−20,−20,20)( - 20 , - 20,20 )(−20,−20,20)

Find the magnitude of the vector described below. Initial point: (3,-1,2) Terminal point: (4,-7,4)

Find the general form of the equation of the plane passing through the three points.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (3,1,6), (-6,-1,6), (3,5,-6)

Find a unit vector in the direction of the vector described below. Initial point: (4,-6,-1) Terminal point: (2,5,-1)

Determine the values of c such that ∥cu∥=9\| c u \| = 9∥cu∥=9 ,where u=i+8j+9k\mathbf { u } = \mathbf { i } + 8 \mathbf { j } + 9 \mathbf { k }u=i+8j+9k .

Find the triple scalar product u · (v × w) for the vectors u=(7,9,4),v=(9,−4,1),w=(8,8,−4)\mathbf { u } = ( 7,9,4 ) , \mathbf { v } = ( 9 , - 4,1 ) , \mathbf { w } = ( 8,8 , - 4 )u=(7,9,4),v=(9,−4,1),w=(8,8,−4)

Use the vectors u and v to find u × (-v). u=7i−j+8kv=6i+6j−k\mathbf { u } = 7 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } \quad \mathbf { v } = 6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }u=7i−j+8kv=6i+6j−k

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (5,5,-2)and (4,2,1)and is perpendicular to the plane -3x - 2y + 4z = 1.

Find the distance between the points. (−5,1,−4),(8,0,−6)( - 5,1 , - 4 ) , ( 8,0 , - 6 )(−5,1,−4),(8,0,−6)

Find the distance between the points. (-2,4,-1,), (-5,0,-5)

Find u × v. u=(−5,−3,5)v=(4,−8,0)\mathbf { u } = ( - 5 , - 3,5 ) \quad \mathbf { v } = ( 4 , - 8,0 )u=(−5,−3,5)v=(4,−8,0)

Determine whether u and v are parallel,orthogonal,or neither. u = (−6,−6,5)( - 6 , - 6,5 )(−6,−6,5) ,v = (8,0,9)( 8,0,9 )(8,0,9)

Find the center and radius of the sphere. x2+y2+z2−14z=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 14 z = 0x2+y2+z2−14z=0

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: (3,0,0)( 3,0,0 )(3,0,0) Perpendicular to: x=3−t,y=2−6t,z=4+tx = 3 - t , y = 2 - 6 t , z = 4 + tx=3−t,y=2−6t,z=4+t

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 the lengths of the sides of the triangle with the indicated vertices. $( 4,0,3 ) , ( 4,2,2 ) , ( 1,0,4 )$

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 7 , - 9,3 )$ ,v = $( - 21 , - 27,9 )$

Find symmetric equations for the line through the point and parallel to the specified vector. (9,-6,-5),parallel to $( 7,2 , - 7 )$

Determine whether the triangle is a right triangle,an isosceles triangle,or neither. $( 8,1,1 ) , ( 3 , - 6 , - 1 ) , ( 1 , - 1 , - 8 )$

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 5 , - 5,5 )$ ,v = $( - 20 , - 20,20 )$

Determine the values of c such that $\| c u \| = 9$ ,where $\mathbf { u } = \mathbf { i } + 8 \mathbf { j } + 9 \mathbf { k }$ .

Find the triple scalar product u · (v × w) for the vectors $\mathbf { u } = ( 7,9,4 ) , \mathbf { v } = ( 9 , - 4,1 ) , \mathbf { w } = ( 8,8 , - 4 )$

Use the vectors u and v to find u × (-v). $\mathbf { u } = 7 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } \quad \mathbf { v } = 6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$

Find the distance between the points. $( - 5,1 , - 4 ) , ( 8,0 , - 6 )$

Find u × v. $\mathbf { u } = ( - 5 , - 3,5 ) \quad \mathbf { v } = ( 4 , - 8,0 )$

Determine whether u and v are parallel,orthogonal,or neither. u = $( - 6 , - 6,5 )$ ,v = $( 8,0,9 )$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 14 z = 0$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: $( 3,0,0 )$ Perpendicular to: $x = 3 - t , y = 2 - 6 t , z = 4 + t$