8.7 Mid Chapter Review
Chapter
Chapter 8
Section
8.7
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}

1.08mins
Q2a

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

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

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?

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.

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}.

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).

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).

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}.

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.

2.12mins
Q9

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

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.

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} .

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.

2.12mins
Q13

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

0.42mins
Q14

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

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}.
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).
0.42mins
Q16b

Determine the parametric equations of each line.

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

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)
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)
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)
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)
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).

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).

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} .

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