Find the angle of intersection of the planes in degrees. Round to a tenth of a degree. \[\begin{array} { l } 3 x - 2 y + 4 z = 3 \\ 5 x - 3 y + 3 z = - 6 \end{array}\]

A) $- 0.8 ^ { \circ }$ B) $69.1 ^ { \circ }$ C) $20.9 ^ { \circ }$ D) $43.1 ^ { \circ }$ E) $0.4 ^ { \circ }$ A) $- 0.8 ^ { \circ }$ B) $69.1 ^ { \circ }$ C) $20.9 ^ { \circ }$ D) $43.1 ^ { \circ }$ E) $0.4 ^ { \circ }$

Exam 10: Analytic Geometry in Three Dimensions

D 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)

4.7/5

(33)

Question 41

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)

4.8/5

(32)

Question 42

Find the midpoint of the line segment joining the points. $( - 3,7 , - 8 ) , ( 8,2,5 )$

(Multiple Choice)

4.8/5

(28)

Question 43

For the points $A ( 4 , - 1 , - 1 ) , B ( - 1 , - 2,4 ) , C ( 5 , - 4,4 ) , D ( 0 , - 5,9 )$ : a. Verify that the points are vertices of a parallelogram. Show all work. b. Find the area of the parallelogram. Show all work. c. Determine whether the parallelogram is a rectangle.

(Essay)

4.9/5

(42)

Question 44

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)

4.9/5

(32)

Question 45

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)

4.8/5

(30)

Question 46

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)

4.9/5

(28)

Question 47

Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \mathbf { v } = 7 \mathbf { i } + 8 \mathbf { j } + 3 \mathbf { k } , \mathbf { w } = 4 \mathbf { i } + 7 \mathbf { j } - 3 \mathbf { k }$

(Multiple Choice)

4.9/5

(47)

Question 48

Find symmetric equations for the line through the point and parallel to the specified vector. Show all your work. $( 1,8,6 )$ , parallel to $\langle 9 , - 6,5 \rangle$

(Essay)

4.8/5

(32)

Question 49

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 2,3,5 \rangle , \mathbf { v } = \langle - 3,3 , - 5 \rangle$

(Multiple Choice)

4.9/5

(28)

Question 50

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , \mathbf { v } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$

(Multiple Choice)

4.8/5

(27)

Question 51

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle - 1,6 , - 3 \rangle , \mathbf { v } = \langle - 3,18 , - 9 \rangle$

(Multiple Choice)

4.7/5

(28)

Question 52

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 0 , - 8 , - 4 \rangle , \mathbf { v } = \langle - 5,0 , - 6 \rangle$ , and $\mathbf { w } = \langle - 17,25 , - 1 \rangle$ . $- 2 \mathbf { u } + 4 \mathbf { v } - 3 \mathbf { z } = \mathbf { w }$

(Multiple Choice)

4.9/5

(46)

Question 53

Write the component form of the vector described below. Initial point: $( - 1 , - 5 , - 6 )$ Terminal point: $( 4 , - 2 , - 4 )$

(Multiple Choice)

5.0/5

(24)

Question 54

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)

4.9/5

(34)

Question 55

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)

4.9/5

(35)

Question 56

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. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$

(Essay)

4.8/5

(35)

Question 57

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . Express your answer in degrees and round to the nearest tenth of a degree. $\mathbf { u } = - 5 \mathbf { i } - 6 \mathbf { j } - \mathbf { k } , \mathbf { v } = 6 \mathbf { i } + \mathbf { j } + \mathbf { k }$

(Multiple Choice)

4.8/5

(43)

Question 58

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+4z=3 5x-3y+3z=-6

(Multiple Choice)

5.0/5

(38)

Question 59

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.] $( - 2 , - 1,4 ) , ( 3 , - 1,2 ) , ( - 6 , - 5 , - 6 )$

(Multiple Choice)

4.9/5

(35)

Question 60

D 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 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 midpoint of the line segment joining the points. $( - 3,7 , - 8 ) , ( 8,2,5 )$

For the points $A ( 4 , - 1 , - 1 ) , B ( - 1 , - 2,4 ) , C ( 5 , - 4,4 ) , D ( 0 , - 5,9 )$ : a. Verify that the points are vertices of a parallelogram. Show all work. b. Find the area of the parallelogram. Show all work. c. Determine whether the parallelogram is a rectangle.

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 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 symmetric equations for the line through the point and parallel to the specified vector. Show all your work. $( 1,8,6 )$ , parallel to $\langle 9 , - 6,5 \rangle$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 2,3,5 \rangle , \mathbf { v } = \langle - 3,3 , - 5 \rangle$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , \mathbf { v } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle - 1,6 , - 3 \rangle , \mathbf { v } = \langle - 3,18 , - 9 \rangle$

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 0 , - 8 , - 4 \rangle , \mathbf { v } = \langle - 5,0 , - 6 \rangle$ , and $\mathbf { w } = \langle - 17,25 , - 1 \rangle$ . $- 2 \mathbf { u } + 4 \mathbf { v } - 3 \mathbf { z } = \mathbf { w }$

Write the component form of the vector described below. Initial point: $( - 1 , - 5 , - 6 )$ Terminal point: $( 4 , - 2 , - 4 )$

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 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 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. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . Express your answer in degrees and round to the nearest tenth of a degree. $\mathbf { u } = - 5 \mathbf { i } - 6 \mathbf { j } - \mathbf { k } , \mathbf { v } = 6 \mathbf { i } + \mathbf { j } + \mathbf { k }$

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+4z=3 5x-3y+3z=-6

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.] $( - 2 , - 1,4 ) , ( 3 , - 1,2 ) , ( - 6 , - 5 , - 6 )$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Limits and an Introduction to Calculus

Filters

Exam 10: Analytic Geometry in Three Dimensions

D 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 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 midpoint of the line segment joining the points. (−3,7,−8),(8,2,5)( - 3,7 , - 8 ) , ( 8,2,5 )(−3,7,−8),(8,2,5)

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 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 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 symmetric equations for the line through the point and parallel to the specified vector. Show all your work. (1,8,6)( 1,8,6 )(1,8,6) , parallel to ⟨9,−6,5⟩\langle 9 , - 6,5 \rangle⟨9,−6,5⟩

Find the area of the parallelogram that has the vectors as adjacent sides. u=⟨2,3,5⟩,v=⟨−3,3,−5⟩\mathbf { u } = \langle 2,3,5 \rangle , \mathbf { v } = \langle - 3,3 , - 5 \rangleu=⟨2,3,5⟩,v=⟨−3,3,−5⟩

Find the area of the parallelogram that has the vectors as adjacent sides. u=−3i−2j+5k,v=i−2j+k\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , \mathbf { v } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k }u=−3i−2j+5k,v=i−2j+k

Determine whether u\mathbf { u }u and v\mathbf { v }v are parallel, orthogonal, or neither. u=⟨−1,6,−3⟩,v=⟨−3,18,−9⟩\mathbf { u } = \langle - 1,6 , - 3 \rangle , \mathbf { v } = \langle - 3,18 , - 9 \rangleu=⟨−1,6,−3⟩,v=⟨−3,18,−9⟩

Write the component form of the vector described below. Initial point: (−1,−5,−6)( - 1 , - 5 , - 6 )(−1,−5,−6) Terminal point: (4,−2,−4)( 4 , - 2 , - 4 )(4,−2,−4)

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 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 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. (−9,6,−1),(−8,3,−2),(3,6,0)( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )(−9,6,−1),(−8,3,−2),(3,6,0)

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+4z=3 5x-3y+3z=-6

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Limits and an Introduction to Calculus

Filters

D 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 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 midpoint of the line segment joining the points. $( - 3,7 , - 8 ) , ( 8,2,5 )$

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 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 symmetric equations for the line through the point and parallel to the specified vector. Show all your work. $( 1,8,6 )$ , parallel to $\langle 9 , - 6,5 \rangle$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 2,3,5 \rangle , \mathbf { v } = \langle - 3,3 , - 5 \rangle$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , \mathbf { v } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle - 1,6 , - 3 \rangle , \mathbf { v } = \langle - 3,18 , - 9 \rangle$

Write the component form of the vector described below. Initial point: $( - 1 , - 5 , - 6 )$ Terminal point: $( 4 , - 2 , - 4 )$

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 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 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. $( - 9,6 , - 1 ) , ( - 8,3 , - 2 ) , ( 3,6,0 )$