8.8 Chapter Test
Chapter
Chapter 8
Section
8.8
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Solutions 28 Videos

A line in two-space has scalar equation 3 x+5 y+9=0 . Which vector is parallel to the line?

A [3,5]

B \displaystyle [3,-5]

C [5,3]

D \displaystyle [-5,3]

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Q1

Which line is the same as

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

\displaystyle \mathbf{A} [x, y]=[2,4]+t[-4,3]

\displaystyle \mathbf{B}[x, y]=[13,-17]+t[8,-6]

\displaystyle \mathbf{C}[x, y]=[-10,20]+t[12,-9]

\displaystyle \mathbf{D}[x, y]=[-11,-4]+t[4,-3]

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Q2

Which plane does not contain the point

\displaystyle \mathrm{P } (10,-3,5) ?

\displaystyle \mathbf{A } 3 x+6 y-2 z-2=0

\displaystyle \mathbf{B } x+y-z-12=0

\displaystyle \mathbf{C } 2 x-2 y-3 z-11=0

\displaystyle \mathbf{D} 4 x+5 y+z-30=0

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Q3

Which does not exist for a line in three-space?

\mathbf{A} vector equation

\mathbf{B} scalar equation

\mathbf{C} parametric equation

\mathbf{D} a point and a direction vector

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Q4

The equation 2 x+5 y=9 can represent

A a scalar equation of a line in two-space

B a scalar equation of a plane in three-space

C neither A nor B

D both A and B

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Q5

Which vector is not normal to the plane

\displaystyle x+2 y-3 z-4=0 ?

\displaystyle \mathbf{A }[2,-3,4]

\displaystyle \mathbf{B }[-1,-2,3]

\displaystyle \mathbf{C }[2,4,-6]

\displaystyle \mathbf{D }[3,6,-9]

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Q6

Which point is not a solution to the equation

\displaystyle 3 x-4 y+z-12=0 ?

\displaystyle \mathbf{A}(-8,-9,0)

\displaystyle \mathbf{B} (4,1,4)

\displaystyle \mathbf{C} (16,10,4)

\displaystyle \mathbf{D} (18,12,2)

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Q7

How many solutions does this system have?

\displaystyle 10 x-7 y=10

\displaystyle 4 x+5 y=12

\mathbf{A} 0

\displaystyle \mathbf{B} 1

\mathbf{C} 12

\mathbf{D} infinitely many

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Q8

Which word best describes the solution to this

system of equations?

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

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

A consistent

B coincident

C inconsistent

D skew

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Q9

Which statement is never true for two planes?

A they intersect at a point

B they intersect at a line

C they are parallel and distinct

D they are coincident

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Q10

A line passes through points \mathrm{A}(1,5,-4) and

\displaystyle \mathrm{B}(2,-9,0)

a) Write vector and parametric equations of the line.

b) Determine two other points on the line.

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Q11

Write the parametric equations of a line perpendicular to 4 x+8 y+7=0 with the same x-intercept as \displaystyle [x, y]=[2,7]+t[-10,3] .

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Q12

Find the parametric equations of the line through the point \mathrm{P}(-6,4,3) , that is perpendicular to both of the lines

\displaystyle [x, y, z]=[0,-10,-2]+s[4,6,-3]

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

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Q13

Write the scalar equation of a plane that is parallel to the x z -plane and contains the line

\displaystyle [x, y, z]=[3,-1,5]+t[4,0,-1]

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Q14

Determine if the lines in each pair intersect. If they intersect, find the intersection point.

\displaystyle 6 x+2 y=5

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

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Q15a

Determine if the lines in each pair intersect. If they intersect, find the intersection point.

\displaystyle 2 x+3 y=21

\displaystyle 4 x-y=7

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Q15b

Determine if the lines in each pair intersect. If they intersect, find the intersection point.

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

[x, y, z]=[7,10,4]+t[1,4,3]

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Q15c

Determine if the lines in each pair intersect. If they intersect, find the intersection point.

\displaystyle [x, y, z]=[3,4,-6]+s[5,2,-2]

\displaystyle [x, y, z]=[1,-4,4]+t[10,4,-4]

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Q15d

Determine whether the line with equation

\displaystyle [x, y, z]=[-3,-6,-11]+k[22,1,-11]

in the plane that contains the points \mathrm{A}(2,5,6) , \mathrm{B}(-7,1,4) , \mathrm{C}(6,-2,-9)

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Q17

Determine if the planes in each set intersect. If so, describe how they intersect.

\displaystyle 2 x+5 y-7 z=31

\displaystyle x-2 y-5 z=-9

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Q18a

Determine if the planes in each set intersect. If so, describe how they intersect.

\displaystyle x+y-z=11

\displaystyle 2 x+3 y+4 z=0

\displaystyle 2 x-2 y+z=4

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Q18b

Determine if the planes in each set intersect. If so, describe how they intersect.

\displaystyle 2 x+y-3 z=2

\displaystyle x-4 y+2 z=5

\displaystyle 4x+2 y-6 z=-12

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Q18c

Determine if the planes in each set intersect. If so, describe how they intersect.

\displaystyle 2 x-3 y-4 z=-16

\displaystyle 11 x-3 y+5 z=47

\displaystyle 5 x+y+7 z=45

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Q18d

A plane passes through the points \mathrm{A}(1,13,2) ,

\displaystyle \mathrm{B}(-2,-6,5)

\displaystyle \mathrm{C}(-1,-1,-3)

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

b) Write the scalar equation of the plane.

c) Determine two other points on the plane.

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Q19

The plane \pi has vector equation

\displaystyle \vec{r}=[2,1,3]+s[1,1,1]+t[2,0,2], s, t \in \mathbb{R}

a) Verify that \pi does not pass through the origin.

b) Find the distance from the origin to \pi .

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Q20

A plane \pi_{1} has equation 2 x-3 y+5 z=1 . Two other planes, \pi_{2} and \pi_{3} , intersect the plane, and each other, as illustrated in the diagram. State possible equations for \pi_{2} and \pi_{3} and justify your reasoning.

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Q21

Given the plane 2 x+3 y+4 z=24 and the lines

\displaystyle [x, y, z]=[0,0,6]+r[3,2,-3]

\displaystyle [x, y, z]=[14,-12,8]+s[1,-2,1]

\displaystyle [x, y, z]=[9,-10,9]+t[1,6,-5]

a) show that each line is contained in the plane

b) find the vertices of the triangle formed by the lines

c) find the perimeter of the triangle formed by the lines

d) find the area of the triangle formed by the lines

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Q22