8.7 Chapter Review
Chapter
Chapter 8
Section
8.7
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Solutions 41 Videos

Write the vector and parametric equations of each line.

\displaystyle \vec{m}=[1,2], \mathrm{P}(-3,2)

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Q1a

Write the vector and parametric equations of each line.

\displaystyle \vec{m}=[6,5,1], \mathrm{P}(-9,0,4)

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Q1b

Write the vector and parametric equations of each line.

parallel to the x -axis with z -intercept 7

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Q1c

Write the vector and parametric equations of each line.

perpendicular to the x y -plane and through

\displaystyle (3,0,-4)

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Q1d

Given each scalar equation, write a vector equation.

\displaystyle 5 x-2 y=9

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Q2a

Given each scalar equation, write a vector equation.

\displaystyle x+7 y=10

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Q2b

Given each scalar equation, write a vector equation.

\displaystyle x=8

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Q2c

Given each scalar equation, write a vector equation.

\displaystyle x-4 y=0

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Q2d

Write the scalar equation for each line.

\displaystyle [x, y]=[1,4]+t[2,7]

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Q3a

Write the scalar equation for each line.

\displaystyle [x, y]=[10,-3]+t[5,-7]

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Q3b

A line is defined by the equation

\displaystyle [x, y, z]=[1,-1,5]+t[3,4,7]

Write the parametric equations for the line.

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Q4a

A line is defined by the equation

\displaystyle [x, y, z]=[1,-1,5]+t[3,4,7]

Does the point (13,15,23) lie on the line?

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Q4b

The vertices of a parallelogram are the origin and points \mathrm{A}(-1,4), \mathrm{B}(3,6) , and \mathrm{C}(7,2) . Write the vector equations of the lines that make up the sides of the parallelogram.

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Q5

A line has the same x -intercept as

\displaystyle [x, y, z]=[-21,8,14]+t[-12,4,7]

and the same y -intercept as

\displaystyle [x, y, z]=[6,-8,12]+s[2,-5,4]

Write the parametric equations of the line.

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Q6

Find three points on each plane.

\displaystyle [x, y, z]=[3,4,-1]+s[1,1,-4]+t|2,-5,3|

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Q7a

Find three points on each plane.

\displaystyle x+2 y-z+12=0

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Q7b

Find three points on each plane.

\displaystyle x=3 k+4 p\\y=-5-2 k+p\\z=2+3 k-2 p

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Q7c

A plane contains the line

\displaystyle [ x, y, z] =[ 2,-9,10]+t[3,-8,7]

the point \mathrm{P}(5,1,3) . Write the vector and parametric equations of the plane.

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Q8

Does \mathrm{P}(-3,4,-5) lie on each plane?

\displaystyle [x, y, z]=[1,-5,6]+s[2,1,3]+t[1,7,1]

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Q9a

Does \mathrm{P}(-3,4,-5) lie on each plane?

\displaystyle 4 x+y-2 z-2=0

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Q9b

Do the points \mathrm{A}(2,1,5), \mathrm{B}(-1,-1,10) , and \mathrm{C}(8,5,-5) define a plane? Explain why or why not.

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Q10

A plane is defined by the equation

\displaystyle x-4 y+2 z=16

a) Find two vectors parallel to the plane.

b) Determine the x -, y -, and z -intercepts.

c) Write the vector and parametric equations of the plane.

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Q11

Write the scalar equation of the plane with

\vec{n}=[1,2,-9]

P(3,-4,0)

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Q12

Write the scalar equation of the plane

[x, y, z]=[5,4,-7]+s[0,1,0]+t[0,0,1] .

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Q13

Write the scalar equation of each plane.

parallel to the y z -plane with x -intercept 4

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Q14a

Write the scalar equation of each plane.

parallel to the vector \vec{a}=[3,-7,1] and to the y -axis, and through (1,2,4)

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Q14b

Determine the number of solutions for each linear system in two-space. If possible, solve each system.

\displaystyle 2 x-5 y=6\\\left\{\begin{array}{l}x=-9+7 t \\ y=-4+3 t\end{array}\right.

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Q15a

Determine the number of solutions for each linear system in two-space. If possible, solve each system.

\displaystyle [x, y]=[9,4]+s[1,1]\\[x, y]=[0,9]+t[3,-4]

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Q15b

Write two other equations that have the same solution as this system of equations.

\displaystyle 3 x-4 y=-14

\displaystyle -x+3 y=18

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Q16

Determine if the lines in each pair intersect.

If so, find the coordinates of the point of It so, intersection

\displaystyle [x, y, z]=[1,5,-2]+s[1,7,-3]

\displaystyle [x, y, z]=[-3,-23,10]+t[1,7,-3]

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Q17a

Determine if the lines in each pair intersect.

If so, find the coordinates of the point of It so, intersection

\displaystyle [x, y, z]=[15,2,-1]+s[4,1,-1]

\displaystyle [x, y, z]=[13,-5,-4]+t[-5,2,3]

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Q17b

Find the distance between these skew lines.

\displaystyle [x, y, z]=[1,0,-1]+s[2,3,-4]

\displaystyle [x, y, z]=[8,1,3]+t[4,-5,1]

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Q18

Determine if each line intersects the plane. If so, state the solution.

\displaystyle 5 x-2 y+4 z=23

\displaystyle [x, y, z]=[-17,7,-6]+t[4,1,-3]

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Q19a

Determine if each line intersects the plane. If so, state the solution.

\displaystyle x+4 y+3 z=11

\displaystyle [x, y, z]=[-1,-9,16]+t[3,3,-5]

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Q19b

Find the distance between point \mathrm{P}(3,-2,0) and the plane 4 x-y+8 z=2 .

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Q20

Find the line of intersection for these two planes.

\displaystyle 3 x+y+z=10

\displaystyle 5 x+4 y-2 z=31

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Q21

How do the planes in the system intersect?

\displaystyle 2 x+5 y+2 z=3

\displaystyle x+2 y-3 z=-11

\displaystyle 2 x+y+5 z=8

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Q22a

How do the planes in the system intersect?

\displaystyle x+3 y+2 z=10

\displaystyle 3 x-5 y+z=1

\displaystyle 6 x+4 y+7 z=-5

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Q22b

How do the planes in the system intersect?

\displaystyle x+3 y-z=-2

\displaystyle 3 x+y+z=14

\displaystyle 5 x+7 y+z=10

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Q22c

Use the normal vectors of the planes to describe each system.

\displaystyle 2 x+5 y+3 z=0

\displaystyle x-3 y+6 z=19

\displaystyle 3 x+2 y+9 z=-7

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Q23a

Use the normal vectors of the planes to describe each system.

\displaystyle 8 x+20 y+16 z=3

\displaystyle 3 x+15 y+12 z=10

\displaystyle 2 x-5 y+4 z=2

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Q23b