Find the area of the parallelogram that has the vectors as adjacent sides. $\begin{array} { l } \mathbf { u } = 2 \mathbf { k } \\ \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { k } \end{array}$

A) $\text { Area } = 5 \text { square units }$ B) $\text { Area } = 7 \text { square units }$ C) $\text { Area } = 4 \text { square units }$ D) $\text { Area } = 8 \text { square units }$ E) $\text { Area } = 6 \text { square units }$ A) $\text { Area } = 5 \text { square units }$ B) $\text { Area } = 7 \text { square units }$ C) $\text { Area } = 4 \text { square units }$ D) $\text { Area } = 8 \text { square units }$ E) $\text { Area } = 6 \text { square units }$

Determine whether the planes are parallel,orthogonal,or neither. $\begin{array} { l } 3 x - z = 3 \\ 7 x + y + 21 z = 7 \end{array}$

A) Neither B) Parallel C) Orthogonal A) Neither B) Parallel C) Orthogonal

Exam 11: Analytic Geometry In Three Dimensions

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line. Point: (7,0,0) Parallel to: $\mathbf { v } = ( 5,6,7 )$

(Multiple Choice)

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Question 41

Find the standard form of the equation of the sphere with the given characteristics. Center: $( 4,3,5 )$ ;radius: 5

(Multiple Choice)

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Question 42

Determine whether the planes are parallel,orthogonal,or neither. x + 2y + z = 6 -2x - 4y - 2z = -10

(Multiple Choice)

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Question 43

Use the vectors u and v to find v · (v × u). $\mathbf { u } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } \quad \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$

(Multiple Choice)

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Question 44

Find u × v. $\mathbf { u } = ( 6 ) \mathbf { i } + ( 3 ) \mathbf { j } + ( 2 ) \mathbf { k } \quad \mathbf { v } = ( 5 ) \mathbf { i } + ( 2 ) \mathbf { j } + ( - 1 ) \mathbf { k }$

(Multiple Choice)

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Question 45

The vector v and its initial point are given.Find the terminal point. v = $( - 1,5 , - 7 )$ Initial point: (5,-7,-2)

(Multiple Choice)

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Question 46

Find the area of the triangle with the given vertices. $A ( - 3 , - 4 , - 9 ) , B ( - 7 , - 9 , - 1 ) , C ( 8,8,6 )$

(Multiple Choice)

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Question 47

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 6 , - 1,7 ) \quad \mathbf { v } = ( 5,5 , - 1 )$

(Multiple Choice)

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Question 48

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = $( 0 , - 8 , - 7 )$ ,v = $( - 7 , - 6 , - 7 )$

(Multiple Choice)

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Question 49

Find the area of the parallelogram that has the vectors as adjacent sides. =2 =2+2

(Multiple Choice)

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Question 50

Use vectors to determine whether the points are collinear. (-3,-3,7), (2,-4,6), (1,-4,5)

(Multiple Choice)

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Question 51

Find the distance between the points. (0,0,0), (-4,9,-8)

(Multiple Choice)

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Question 52

Find a set of parametric equations of the line. Passes through $( 3 , - 4 , - 3 )$ and is parallel to $\mathbf { v } = ( 6 , - 6,3 )$ .

(Multiple Choice)

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Question 53

Determine whether the planes are parallel,orthogonal,or neither. 3x-z=3 7x+y+21z=7

(Multiple Choice)

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Question 54

Find the magnitude of v. Initial point: $( 1 , - 5,6 )$ Terminal point: $( 1,0 , - 1 )$

(Multiple Choice)

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Question 55

Find a set of parametric equations of the line. Passes through $( 4,6,5 )$ and is perpendicular to $3 x + 6 y - z = 6$ .

(Multiple Choice)

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Question 56

Find the general form of the equation of the plane with the given characteristics. Passes through $( 7,4,6 )$ and is parallel to the yz-plane

(Multiple Choice)

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Question 57

Find the magnitude of the vector v. v = $( - 6 , - 7 , - 9 )$

(Multiple Choice)

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Question 58

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: $( 2 , - 5,3 )$ Parallel to: $v = ( 4 , - 3 , - 6 )$

(Multiple Choice)

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(27)

Question 59

Find the triple scalar product $\mathbf { u } ^ { * } ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = ( 7,4 , - 5 ) , \mathbf { v } = ( 6,6 , - 1 ) , \mathbf { w } = ( - 5,4,4 )$

(Multiple Choice)

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Question 60

Find a set of symmetric equations for the line through the point and parallel to the specified vector or line. Point: (7,0,0) Parallel to: $\mathbf { v } = ( 5,6,7 )$

Find the standard form of the equation of the sphere with the given characteristics. Center: $( 4,3,5 )$ ;radius: 5

Determine whether the planes are parallel,orthogonal,or neither. x + 2y + z = 6 -2x - 4y - 2z = -10

Use the vectors u and v to find v · (v × u). $\mathbf { u } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } \quad \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$

Find u × v. $\mathbf { u } = ( 6 ) \mathbf { i } + ( 3 ) \mathbf { j } + ( 2 ) \mathbf { k } \quad \mathbf { v } = ( 5 ) \mathbf { i } + ( 2 ) \mathbf { j } + ( - 1 ) \mathbf { k }$

The vector v and its initial point are given.Find the terminal point. v = $( - 1,5 , - 7 )$ Initial point: (5,-7,-2)

Find the area of the triangle with the given vertices. $A ( - 3 , - 4 , - 9 ) , B ( - 7 , - 9 , - 1 ) , C ( 8,8,6 )$

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 6 , - 1,7 ) \quad \mathbf { v } = ( 5,5 , - 1 )$

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = $( 0 , - 8 , - 7 )$ ,v = $( - 7 , - 6 , - 7 )$

Find the area of the parallelogram that has the vectors as adjacent sides. =2 =2+2

Use vectors to determine whether the points are collinear. (-3,-3,7), (2,-4,6), (1,-4,5)

Find the distance between the points. (0,0,0), (-4,9,-8)

Find a set of parametric equations of the line. Passes through $( 3 , - 4 , - 3 )$ and is parallel to $\mathbf { v } = ( 6 , - 6,3 )$ .

Determine whether the planes are parallel,orthogonal,or neither. 3x-z=3 7x+y+21z=7

Find the magnitude of v. Initial point: $( 1 , - 5,6 )$ Terminal point: $( 1,0 , - 1 )$

Find a set of parametric equations of the line. Passes through $( 4,6,5 )$ and is perpendicular to $3 x + 6 y - z = 6$ .

Find the general form of the equation of the plane with the given characteristics. Passes through $( 7,4,6 )$ and is parallel to the yz-plane

Find the magnitude of the vector v. v = $( - 6 , - 7 , - 9 )$

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: $( 2 , - 5,3 )$ Parallel to: $v = ( 4 , - 3 , - 6 )$

Find the triple scalar product $\mathbf { u } ^ { * } ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = ( 7,4 , - 5 ) , \mathbf { v } = ( 6,6 , - 1 ) , \mathbf { w } = ( - 5,4,4 )$

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 for the line through the point and parallel to the specified vector or line. Point: (7,0,0) Parallel to: v=(5,6,7)\mathbf { v } = ( 5,6,7 )v=(5,6,7)

Find the standard form of the equation of the sphere with the given characteristics. Center: (4,3,5)( 4,3,5 )(4,3,5) ;radius: 5

Determine whether the planes are parallel,orthogonal,or neither. x + 2y + z = 6 -2x - 4y - 2z = -10

Use the vectors u and v to find v · (v × u). u=3i−j+4kv=2i+2j−k\mathbf { u } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } \quad \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }u=3i−j+4kv=2i+2j−k

Find u × v. u=(6)i+(3)j+(2)kv=(5)i+(2)j+(−1)k\mathbf { u } = ( 6 ) \mathbf { i } + ( 3 ) \mathbf { j } + ( 2 ) \mathbf { k } \quad \mathbf { v } = ( 5 ) \mathbf { i } + ( 2 ) \mathbf { j } + ( - 1 ) \mathbf { k }u=(6)i+(3)j+(2)kv=(5)i+(2)j+(−1)k

The vector v and its initial point are given.Find the terminal point. v = (−1,5,−7)( - 1,5 , - 7 )(−1,5,−7) Initial point: (5,-7,-2)

Find the area of the triangle with the given vertices. A(−3,−4,−9),B(−7,−9,−1),C(8,8,6)A ( - 3 , - 4 , - 9 ) , B ( - 7 , - 9 , - 1 ) , C ( 8,8,6 )A(−3,−4,−9),B(−7,−9,−1),C(8,8,6)

Find u × v and show that it is orthogonal to both u and v. u=(6,−1,7)v=(5,5,−1)\mathbf { u } = ( 6 , - 1,7 ) \quad \mathbf { v } = ( 5,5 , - 1 )u=(6,−1,7)v=(5,5,−1)

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = (0,−8,−7)( 0 , - 8 , - 7 )(0,−8,−7) ,v = (−7,−6,−7)( - 7 , - 6 , - 7 )(−7,−6,−7)

Find the area of the parallelogram that has the vectors as adjacent sides. =2 =2+2

Use vectors to determine whether the points are collinear. (-3,-3,7), (2,-4,6), (1,-4,5)

Find the distance between the points. (0,0,0), (-4,9,-8)

Find a set of parametric equations of the line. Passes through (3,−4,−3)( 3 , - 4 , - 3 )(3,−4,−3) and is parallel to v=(6,−6,3)\mathbf { v } = ( 6 , - 6,3 )v=(6,−6,3) .

Determine whether the planes are parallel,orthogonal,or neither. 3x-z=3 7x+y+21z=7

Find the magnitude of v. Initial point: (1,−5,6)( 1 , - 5,6 )(1,−5,6) Terminal point: (1,0,−1)( 1,0 , - 1 )(1,0,−1)

Find a set of parametric equations of the line. Passes through (4,6,5)( 4,6,5 )(4,6,5) and is perpendicular to 3x+6y−z=63 x + 6 y - z = 63x+6y−z=6 .

Find the general form of the equation of the plane with the given characteristics. Passes through (7,4,6)( 7,4,6 )(7,4,6) and is parallel to the yz-plane

Find the magnitude of the vector v. v = (−6,−7,−9)( - 6 , - 7 , - 9 )(−6,−7,−9)

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: (2,−5,3)( 2 , - 5,3 )(2,−5,3) Parallel to: v=(4,−3,−6)v = ( 4 , - 3 , - 6 )v=(4,−3,−6)

Find the triple scalar product u∗(v×w)\mathbf { u } ^ { * } ( \mathbf { v } \times \mathbf { w } )u∗(v×w) for the vectors u=(7,4,−5),v=(6,6,−1),w=(−5,4,4)\mathbf { u } = ( 7,4 , - 5 ) , \mathbf { v } = ( 6,6 , - 1 ) , \mathbf { w } = ( - 5,4,4 )u=(7,4,−5),v=(6,6,−1),w=(−5,4,4)

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 for the line through the point and parallel to the specified vector or line. Point: (7,0,0) Parallel to: $\mathbf { v } = ( 5,6,7 )$

Find the standard form of the equation of the sphere with the given characteristics. Center: $( 4,3,5 )$ ;radius: 5

Use the vectors u and v to find v · (v × u). $\mathbf { u } = 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } \quad \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$

Find u × v. $\mathbf { u } = ( 6 ) \mathbf { i } + ( 3 ) \mathbf { j } + ( 2 ) \mathbf { k } \quad \mathbf { v } = ( 5 ) \mathbf { i } + ( 2 ) \mathbf { j } + ( - 1 ) \mathbf { k }$

The vector v and its initial point are given.Find the terminal point. v = $( - 1,5 , - 7 )$ Initial point: (5,-7,-2)

Find the area of the triangle with the given vertices. $A ( - 3 , - 4 , - 9 ) , B ( - 7 , - 9 , - 1 ) , C ( 8,8,6 )$

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 6 , - 1,7 ) \quad \mathbf { v } = ( 5,5 , - 1 )$

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = $( 0 , - 8 , - 7 )$ ,v = $( - 7 , - 6 , - 7 )$

Find a set of parametric equations of the line. Passes through $( 3 , - 4 , - 3 )$ and is parallel to $\mathbf { v } = ( 6 , - 6,3 )$ .

Find the magnitude of v. Initial point: $( 1 , - 5,6 )$ Terminal point: $( 1,0 , - 1 )$

Find a set of parametric equations of the line. Passes through $( 4,6,5 )$ and is perpendicular to $3 x + 6 y - z = 6$ .

Find the general form of the equation of the plane with the given characteristics. Passes through $( 7,4,6 )$ and is parallel to the yz-plane

Find the magnitude of the vector v. v = $( - 6 , - 7 , - 9 )$

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: $( 2 , - 5,3 )$ Parallel to: $v = ( 4 , - 3 , - 6 )$

Find the triple scalar product $\mathbf { u } ^ { * } ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = ( 7,4 , - 5 ) , \mathbf { v } = ( 6,6 , - 1 ) , \mathbf { w } = ( - 5,4,4 )$