Chapter Review
Chapter
Chapter 8
Section
Chapter Review
You need to sign up or log in to purchase.
Solutions 39 Videos

Determine vector and parametric equations of the plane that contains the points A(1, 2, -1), B(2, 1,1), and C(3, 1, 4).

Buy to View
1.03mins
Q1

Find the Cartesian equation of the plane that's defined by A(1, 2, -1), B(2, 1,1), and C(3, 1, 4).

Buy to View
1.11mins
Q2

a. Determine the vector, parametric, and symmetric equations of the line passing through points A(-3, 2, 8) and B(4, 3, 9).

b. Determine the vector and parametric equations of the plane containing the points A(-2, 2, 8), B(4, 3, 9), and C(-2, -1, 3).

c. Explain why a symmetric equation cannot exist for a plane.

Buy to View
1.03mins
Q3

Determine the vector, parametric, and symmetric equations of the line passing through the points A(7, 1, -2) and perpendicular to the plane with equation 2x - 3y + z -1 =0

Buy to View
1.01mins
Q4

Determine the Cartesian equation of each of the following planes:

through the point P(0, 1, -2), with normal \vec{n} = (-1, 3, 3)

Buy to View
0.38mins
Q5a

Determine the Cartesian equation of each of the following planes:

through the points (3, 0, 1) and (0, 1, -1) and perpendicular to the plane with equation x - y - z + 1 = 0

Buy to View
2.12mins
Q5b

Determine the Cartesian equation of each of the following planes:

through the points (1, 2, 1) and (2, 1, 4), and parallel to the x-axis.

Buy to View
1.12mins
Q5c

Determine the Cartesian equation of the plane that passes through the origin and contains the lie \vec{r} = (3, 7, 1) + t (2, 2, 3), t\in \mathbb{R}.

Buy to View
1.19mins
Q6

Find the vector and parametric equations of the plane that is parallel to the yz-plane and contains the point A(-1, 2, 1).

Buy to View
0.53mins
Q7

Determine the Cartesian equation of the lane that contains the line  \displaystyle \vec{r} = (2, 3, 2) + t(1, 1, 4), t\in \mathbb{R} , and the point (4, -3, 2).

Buy to View
2.08mins
Q8

Determine the Cartesian equation of the plane that contains the following lines:

 \displaystyle L_1: \vec{r} = (4, 4, 5) + t(5, -4, 6), t\in \mathbb{R}  and

 \displaystyle L_2: (4, 4, 5) + s(2, -3, -4), s\in \mathbb{R} 

Buy to View
2.00mins
Q9

Determine an equation for the line that is perpendicular to the plane 3x - 2y + z = 1 passing through (2, ,3 -3). give your answer in vector, parametric, and symmetric form.

Buy to View
1.10mins
Q10

A plane has 3x + 2y -z + 6= 0 as its Cartesian equation. Determine the vector and parametric equations of this plane.

Buy to View
1.52mins
Q11

Determine an equation for the line that has the same x- and z-intercepts as the plane with equation 2x + 5y -z + 7 =0. Give your answer in vector, parametric, and symmetric form.

Buy to View
2.01mins
Q12

Determine the vector, parametric, and Cartesian forms of the equation of the plane containing the lines L_1: \vec{r_1} = (3, -4, 1) + s(1, -3, -5) and L_2: \vec{r_2} = (7, -1, 0) +t (2, -6, -10) where s, t \in \mathbb{R}

Buy to View
3.54mins
Q13

Sketch the plane: 2x + 3y - 6z - 12 = 0

Buy to View
2.07mins
Q14a

Sketch the plane: 2x + 3y - 12 = 0

Buy to View
1.34mins
Q14b

Sketch the plane: x -3z - 6 = 0

Buy to View
1.40mins
Q14c

Sketch the plane: y - 2z - 4 = 0

Buy to View
1.39mins
Q14d

Sketch the plane: 2x + 3y - 6z = 0

Buy to View
4.15mins
Q14e

Determine the vector, parametric, and Cartesian equations of each of the following planes.

passing through the points P(1, -2, 5) and Q(3, 1, 2) and parallel to the line with equation L:\vec{r}= 2t\vec{i} + (4t _3)\vec{j} + (t + 1)\vec{k}, t\in \mathbb{R}.

Buy to View
3.52mins
Q15a

Determine the vector, parametric, and Cartesian equations of each of the following planes.

containing the point A(1, 1, 2) and perpendicular to the line joining the points B(2, 1, -6) and C(-2, 1, 5)

Buy to View
3.16mins
Q15b

Determine the vector, parametric, and Cartesian equations of each of the following planes.

passing through the points (4, 1, -1) and (5, -2, 4) and parallel to the z-axis.

Buy to View
2.07mins
Q15c

Determine the vector, parametric, and Cartesian equations of each of the following planes.

passing through the points (1, ,3 -5), (2, 6, 4), and (3, -3, 3)

Buy to View
2.53mins
Q15d

Show that L_1: \vec{r} = (1, 2, 3) + s(-3, 5, 21)+t(0, 1, 3), s, t \in \mathbb{R}, and L_2: \vec{r} = (1, -1. -6) + u(1, 1, 1) + v(2, 5, 11), u,v, \in \mathbb{R}, are equations for the same plane.

Buy to View
2.18mins
Q16

The two lines L_1: \vec{r_1} = (-1, 1, 0) + s(2, 1, -1) s\in \mathbb{R}, and L_2: \vec{r_2} = (2, 1, 2) + t(2, 1, -1), t \in \mathbb{R}, are parallel but do no coincide. The point A(5, ,4 -3) is on L_1. Determine the coordinates of a point B on L_2 such that \vec{AB} is perpendicular to L_2.

Buy to View
3.18mins
Q17

Write a brief description of each plane.

2x - 3y = 6

Buy to View
1.04mins
Q18a

Write a brief description of each plane.

x -3z = 6

Buy to View
0.48mins
Q18b

Write a brief description of each plane.

2y - z = 6

Buy to View
1.13mins
Q18c

Which of the following points lies on the line x = 2t, y = 3+ t, z = 1 + t?

A(2, 4, 2), B(-2, 2,1), C(4, 5, 2), D(6, 6, 2)

Buy to View
1.00mins
Q19a

If the point (a, b, -3) lies on the line, x = 2t, y = 3+ t, z = 1 + t, determine the values of a and b.

Buy to View
0.43mins
Q19b

Calculate the acute angle that is formed by the intersection of each pair of lines.

L_1: \frac{x -1}{1} = \frac{y-3}{5} and L_2: \frac{x - 2}{2} = \frac{1- y}{3}

Buy to View
1.04mins
Q20a

Calculate the acute angle that is formed by the intersection of each pair of lines.

y = 4x + 2 and y = -x + 3

Buy to View
1.16mins
Q20b

Calculate the acute angle that is formed by the intersection of each pair of lines.

L_1: x =-1 + 3t, y = 1+ 4t, z =-2t and L_2: x = -1 + 2s, y = 3s, z = -7 + 2

Buy to View
1.20mins
Q20c

Calculate the acute angle that is formed by the intersection of each pair of lines.

L_1: (x, y, z) = (4, 7, -1) + t(4, 8, -4) and L_2: (1, 5, 4) + s(-1, 2, 3)

Buy to View
0.49mins
Q20d

Calculate the acute angle that is formed by the intersection of each pair of planes.

2x + 3y -z + 9 = 0  and x + 2y + 4 = 0

Buy to View
1.02mins
Q21a

Calculate the acute angle that is formed by the intersection of each pair of planes.

x - y - z - 1 = 0 and 2x + 3y - z + 4 = 0

Buy to View
0.42mins
Q21b

a. Which of the following lines is parallel to the plane 4x + y - z - 10 = 0?
i. \vec{r} = (3, 0, 2) + t(1, -2, 2) ii. x = -3t, y = -5 + 2t, z = -10t iii. \frac{x - 1}{4} = \frac{y + 6}{-1} =\frac{z}{1}

b. do any of these lines lie in the lane in part a.?

Buy to View
2.17mins
Q22

Explain why any plane with a vector equation of the form (x, y, z) = (a, b, c) + s(d, e, f)+ t(a, b, c) will always pass through the origin.

Buy to View
3.44mins
Q26