8.7 Mid Chapter Review
Chapter
Chapter 8
Section
8.7
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Solutions 27 Videos

Find x- and y-intercepts for each of the following lines:

\vec{r} = (3, 1) + t(-3, 5), t\in \mathbb{R}

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1.08mins
Q2a

Find x- and y-intercepts for each of the following lines:

x = -6 + 2s, y = 3 - 2s, s\in \mathbb{R}

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1.00mins
Q2b

Two lines L_1: \vec{r} = (5, 3) + p(-4, 7), p \in \mathbb{R} and L_2: \vec{r} = (5,3) + q(2, 1), q \in \mathbb{R}, intersect at the point with coordinates (5, 3). What is the angle between L_1 and L_2?

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1.19mins
Q3

Determine the angle that the line with equation \vec{r} = t(4, -5), t\in \mathbb{R}, makes with the x-axis and y-axis.

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1.37mins
Q4

Determine a Cartesian equation for the line that passes through the point (4, -3) and is perpendicular to the line \vec{r} = (2, -3) + t(5, -7), t\in \mathbb{R}.

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1.20mins
Q5

Determine an equation in symmetric form of a line parallel to \displaystyle \frac{x -3}{3} = \frac{y -5}{-4} = \frac{z+7}{4} and passing through (0, 0, 2).

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0.37mins
Q6

Determine parametric equations of the line passing through (1, 2, 5) and parallel to the line passing through K(2, 4, 5) and L(3, -5, 6).

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0.47mins
Q7

Determine direction angles (the angles the direction vector makes with the x-axis, y-axis, and z-axis) for the line with parametric equations x = 5 + 2t, y = 12-8t, z= 5+ 7t, t\in \mathbb{R}.

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2.24mins
Q8

Determine an equation in symmetric form for the line passing through P(3, -4, 6) and having direction angles 60^o, 90^o, and 30^o.

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2.12mins
Q9

Write an equation in parametric form for each of the three coordinates axes in \mathbb{R^3}.

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0.31mins
Q10

The two lines with equations \vec{r} = (1, 2, -4) + t(k + 1, 3k + 1, k -3), t \in \mathbb{R}$, and$x = 2 -3s, y = 1 -10s, z = 3 - 5s, s\in \mathbb{R}$`, are given.

a. Determine a value for k if these lines are parallel.

b. Determine a value for k if these lines are perpendicular.

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3.46mins
Q11

Determine the perimeter and area of the triangle whose vertices are the origin and the x- and y- intercepts of the line \displaystyle \frac{x -6}{3} = \frac{y + 8}{-2} .

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2.30mins
Q12

The Cartesian equation of a line is given by 3x + 4y - 24 = 0.

a. Determine a vector equation for this line.

b. Determine the parametric equations of this line.

c. Determine the acute angle that this line makes with the x-axis.

d. Determine a vector equation of the line that is perpendicular to the given line and passes through the origin.

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2.12mins
Q13

Determine the scalar, vector, and parametric equations of the line that passes through points A(-4, 6) and B(8, 4).

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0.42mins
Q14

Determine a unit vector normal to the line defined by the parametric equations x =1 + 2t and y = -5 - 4t.

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1.09mins
Q15

Determine the parametric equations of each line.

  • the line that passes through (-5, 10) and has a slope of -\frac{2}{3}.
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0.27mins
Q16a

Determine the parametric equations of each line.

  • the line that passes through (1, -1) and is perpendicular to the line (x, y) = (4, -6) + t(2, -2).
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0.42mins
Q16b

Determine the parametric equations of each line.

  • the line that passes through (0, 7) and (0, 10).
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0.43mins
Q16c

Given the line (x, y, z) = (12, -8, -4)+ t(-3, 4, 2),

a. determine the intersections with the coordinate planes, if any

b. determine the intercepts with the coordinate axes, if any

c. graph the line in an x-,y-,z-coordinate system.

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3.33mins
Q17

For each of the following, determine vector, parametric, and, if possible, symmetric equations of the line that passes through P_0 nas has direction vector \vec{d}.

  • P_0 = (1, -2, 8), \vec{d} = (-5, -2, 1)
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0.42mins
Q18a

For each of the following, determine vector, parametric, and, if possible, symmetric equations of the line that passes through P_0 nas has direction vector \vec{d}.

  • P_0 = (3, 6, 9), \vec{d} = (2, 4, 6)
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0.43mins
Q18b

For each of the following, determine vector, parametric, and, if possible, symmetric equations of the line that passes through P_0 nas has direction vector \vec{d}.

  • P_0 = (0, 0, 6), \vec{d} = (-1, 5, 1)
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0.32mins
Q18c

For each of the following, determine vector, parametric, and, if possible, symmetric equations of the line that passes through P_0 nas has direction vector \vec{d}.

  • P_0 = (2, 0, 0), \vec{d} = (0, 0, -2)
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0.35mins
Q18d

Determine a vector equation of the line that passes through the origin and is parallel to the line through the points (-4 5, 6) and (6, -5, 4).

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0.50mins
Q19

Determine the parametric equations of the line through (0, -8, 1), and which passes through the midpoint of the segment joining (0, -8, 1) and which passes through the midpoint of the segment joining (2, 6, 10) and (-4, 4, -8).

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2.17mins
Q20

The symmetric equations of two lines are given. Show that these lines are parallel.

\displaystyle L_1: \frac{x - 2}{1} = \frac{y + 3}{3} = \frac{z - 4}{-5} and

\displaystyle L_2: \frac{x +1}{-3} = \frac{y -2}{-9} = \frac{z + 1}{15} .

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1.06mins
Q21

Does the point D(7, -1, 8) lie on the line with symmetric equations \displaystyle \frac{x - 4}{3} = \frac{y + 2}{1} = \frac{z - 6}{2} ? Explain

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1.09mins
Q22