8.7 Chapter Review
Chapter
Chapter 8
Section
8.7
Solutions 41 Videos

Write the vector and parametric equations of each line.

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

Q1a

Write the vector and parametric equations of each line.

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

Q1b

Write the vector and parametric equations of each line.

parallel to the  x  -axis with  z  -intercept 7

Q1c

Write the vector and parametric equations of each line.

perpendicular to the  x y  -plane and through

\displaystyle (3,0,-4) 

Q1d

Given each scalar equation, write a vector equation.

\displaystyle 5 x-2 y=9 

Q2a

Given each scalar equation, write a vector equation.

\displaystyle x+7 y=10 

Q2b

Given each scalar equation, write a vector equation.

\displaystyle x=8 

Q2c

Given each scalar equation, write a vector equation.

\displaystyle x-4 y=0 

Q2d

Write the scalar equation for each line.

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

Q3a

Write the scalar equation for each line.

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

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.

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?

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.

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.

Q6

Find three points on each plane.

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

Q7a

Find three points on each plane.

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

Q7b

Find three points on each plane.

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

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.

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] 

Q9a

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

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

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.

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.

Q11

Write the scalar equation of the plane with

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

P(3,-4,0)

Q12

Write the scalar equation of the plane

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

Q13

Write the scalar equation of each plane.

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

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)

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. 

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] 

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 

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] 

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] 

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] 

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] 

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] 

Q19b

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

Q20

Find the line of intersection for these two planes.

\displaystyle 3 x+y+z=10 

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

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 

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 

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 

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 

\displaystyle 8 x+20 y+16 z=3 
\displaystyle 3 x+15 y+12 z=10 
\displaystyle 2 x-5 y+4 z=2