Find the angle between the two planes in degrees. Round to a tenth of a degree. \[\begin{array} { l } x + 3 y + 6 z = 4 \\ x - 6 y - z = 4 \end{array}\]

A) $9.4 ^ { \circ }$ B) $33.4 ^ { \circ }$ C) $123.4 ^ { \circ }$ D) $28.8 ^ { \circ }$ E) $2.2 ^ { \circ }$ A) $9.4 ^ { \circ }$ B) $33.4 ^ { \circ }$ C) $123.4 ^ { \circ }$ D) $28.8 ^ { \circ }$ E) $2.2 ^ { \circ }$

Find the distance between the point and the plane. \[\begin{array} { l } ( - 2 , - 1 , - 4 ) \\ - 3 x - 2 y - z = 1 \end{array}\]

A) 11 B) $\frac { 11 } { 14 }$ C) $\frac { 13 } { \sqrt { 14 } }$ D) 0 E) $\frac { 11 } { \sqrt { 14 } }$ A) 11 B) $\frac { 11 } { 14 }$ C) $\frac { 13 } { \sqrt { 14 } }$ D) 0 E) $\frac { 11 } { \sqrt { 14 } }$

Exam 10: Analytic Geometry in Three Dimensions

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangle$ , and $\mathbf { w } = \langle - 13,34 , - 10 \rangle$ . $- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }$

(Multiple Choice)

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Question 21

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangle$

(Multiple Choice)

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Question 22

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.] $( - 1 , - 4 , - 2 ) , ( - 6,5,4 ) , ( 1 , - 2 , - 6 )$

(Multiple Choice)

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Question 23

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$

(Multiple Choice)

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Question 24

Find the angle between the two planes in degrees. Round to a tenth of a degree. x+3y+6z=4 x-6y-z=4

(Multiple Choice)

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Question 25

The weight of a crate is 300 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 $A = ( 0,0 , - 130 )$ , point $B = ( 90,0,0 )$ , point $C = ( - 40,40,0 )$ , point $D = ( 0 , - 180,0 )$

(Multiple Choice)

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Question 26

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangle$

(Multiple Choice)

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Question 27

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 9,8 , - 2 ) , ( - 3 , - 6,4 )$

(Essay)

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Question 28

Find the coordinates of the point located three units in front of the $y z$ -plane, eight units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

(Multiple Choice)

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Question 29

Find the torque on the crankshaft $\mathbf { V }$ using the data shown in the figure. Round to the nearest tenth of a foot-pound. $Find the torque on the crankshaft \mathbf { V } using the data shown in the figure. Round to the nearest tenth of a foot-pound. \begin{array} { l } \| \mathbf { V } \| = 1.6 \mathrm { ft } \\ \| \mathbf { F } \| = 20 \mathrm { lb } \\ \theta = 60 ^ { \circ } \end{array}$ \|\|=1.6 \|\|=20 \theta=6

(Multiple Choice)

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Question 30

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( - 4,4,3 )$ and $( 2,7 , - 7 )$ and is perpendicular to the plane $2 x + 3 y + 3 z = 1$ .

(Multiple Choice)

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Question 31

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$ .

(Multiple Choice)

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Question 32

Determine the values of $c$ such that $\| c \mathbf { u } \| = 2$ , where $\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$ .

(Multiple Choice)

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Question 33

Find the distance between the points. $( 1,9,3 ) , ( 5 , - 6,8 )$

(Multiple Choice)

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Question 34

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }$

(Multiple Choice)

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Question 35

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )$

(Essay)

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Question 36

Find the distance between the point and the plane. (-2,-1,-4) -3x-2y-z=1

(Multiple Choice)

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Question 37

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )$

(Essay)

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Question 38

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

(Multiple Choice)

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Question 39

Find the coordinates of the point located five units in front of the $y z$ -plane, seven units to the right of the $x z$ -plane, and nine units below the $x y$ -plane.

(Multiple Choice)

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Question 40

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 5 , - 8,2 \rangle , \mathbf { v } = \langle - 6 , - 2,9 \rangle$ , and $\mathbf { w } = \langle - 13,34 , - 10 \rangle$ . $- 3 \mathbf { u } - 2 \mathbf { v } - 2 \mathbf { z } = \mathbf { w }$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangle$

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.] $( - 1 , - 4 , - 2 ) , ( - 6,5,4 ) , ( 1 , - 2 , - 6 )$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$

Find the angle between the two planes in degrees. Round to a tenth of a degree. x+3y+6z=4 x-6y-z=4

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangle$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 9,8 , - 2 ) , ( - 3 , - 6,4 )$

Find the coordinates of the point located three units in front of the $y z$ -plane, eight units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( - 4,4,3 )$ and $( 2,7 , - 7 )$ and is perpendicular to the plane $2 x + 3 y + 3 z = 1$ .

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$ .

Determine the values of $c$ such that $\| c \mathbf { u } \| = 2$ , where $\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$ .

Find the distance between the points. $( 1,9,3 ) , ( 5 , - 6,8 )$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }$

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )$

Find the distance between the point and the plane. (-2,-1,-4) -3x-2y-z=1

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )$

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

Find the coordinates of the point located five units in front of the $y z$ -plane, seven units to the right of the $x z$ -plane, and nine units below the $x y$ -plane.

Functions and Their Graphs

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Exam 10: Analytic Geometry in Three Dimensions

Find the area of the parallelogram that has the vectors as adjacent sides. u=⟨4,2,−5⟩,v=⟨−4,1,2⟩\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangleu=⟨4,2,−5⟩,v=⟨−4,1,2⟩

Find the dot product of u\mathbf { u }u and v\mathbf { v }v . u=9i−5j−9k,v=−2i+6j+5k\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }u=9i−5j−9k,v=−2i+6j+5k

Find the angle between the two planes in degrees. Round to a tenth of a degree. x+3y+6z=4 x-6y-z=4

Determine whether u\mathbf { u }u and v\mathbf { v }v are parallel, orthogonal, or neither. u=⟨5,6,−8⟩,v=⟨10,12,−16⟩\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangleu=⟨5,6,−8⟩,v=⟨10,12,−16⟩

Find a set of parametric equations for the line that passes through the given points. Show all your work. (9,8,−2),(−3,−6,4)( 9,8 , - 2 ) , ( - 3 , - 6,4 )(9,8,−2),(−3,−6,4)

Find the coordinates of the point located three units in front of the yzy zyz -plane, eight units to the right of the xzx zxz -plane, and five units above the xyx yxy -plane.

Find the torque on the crankshaft V\mathbf { V }V using the data shown in the figure. Round to the nearest tenth of a foot-pound. \|\|=1.6 \|\|=20 \theta=6

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (−4,4,3)( - 4,4,3 )(−4,4,3) and (2,7,−7)( 2,7 , - 7 )(2,7,−7) and is perpendicular to the plane 2x+3y+3z=12 x + 3 y + 3 z = 12x+3y+3z=1 .

Find a set of symmetric equations of the line that passes through the points (5,0,5)( 5,0,5 )(5,0,5) and (7,8,−6)( 7,8 , - 6 )(7,8,−6) .

Determine the values of ccc such that ∥cu∥=2\| c \mathbf { u } \| = 2∥cu∥=2 , where u=3i−4j+2k\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }u=3i−4j+2k .

Find the distance between the points. (1,9,3),(5,−6,8)( 1,9,3 ) , ( 5 , - 6,8 )(1,9,3),(5,−6,8)

Find the area of the parallelogram that has the vectors as adjacent sides. u=2i+j+k,v=5i−5j+5k\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }u=2i+j+k,v=5i−5j+5k

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. (8,−4,4),(9,5,3),(−3,6,0)( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )(8,−4,4),(9,5,3),(−3,6,0)

Find the distance between the point and the plane. (-2,-1,-4) -3x-2y-z=1

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. (−6,−3,4),(−7,−1,3),(6,4,0)( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )(−6,−3,4),(−7,−1,3),(6,4,0)

Find the area of the parallelogram that has the vectors as adjacent sides. u=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩\mathbf { u } = \langle - 4 , - 4 , - 1 \rangle , \mathbf { v } = \langle - 1,4 , - 4 \rangleu=⟨−4,−4,−1⟩,v=⟨−1,4,−4⟩

Find the coordinates of the point located five units in front of the yzy zyz -plane, seven units to the right of the xzx zxz -plane, and nine units below the xyx yxy -plane.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

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Analytic Trigonometry

Additional Topics in Trigonometry

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Topics in Analytic Geometry

Calculus Practice Problems

Filters

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4,2 , - 5 \rangle , \mathbf { v } = \langle - 4,1,2 \rangle$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = 9 \mathbf { i } - 5 \mathbf { j } - 9 \mathbf { k } , \mathbf { v } = - 2 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k }$

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangle$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 9,8 , - 2 ) , ( - 3 , - 6,4 )$

Find the coordinates of the point located three units in front of the $y z$ -plane, eight units to the right of the $x z$ -plane, and five units above the $x y$ -plane.

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( - 4,4,3 )$ and $( 2,7 , - 7 )$ and is perpendicular to the plane $2 x + 3 y + 3 z = 1$ .

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$ .

Determine the values of $c$ such that $\| c \mathbf { u } \| = 2$ , where $\mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$ .

Find the distance between the points. $( 1,9,3 ) , ( 5 , - 6,8 )$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { v } = 5 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }$

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( 8 , - 4,4 ) , ( 9,5,3 ) , ( - 3,6,0 )$

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. $( - 6 , - 3,4 ) , ( - 7 , - 1,3 ) , ( 6,4,0 )$

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

Find the coordinates of the point located five units in front of the $y z$ -plane, seven units to the right of the $x z$ -plane, and nine units below the $x y$ -plane.