Chapter Review
Chapter
Chapter 8
Section
Chapter Review
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).

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).

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.

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

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)

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

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.

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}.

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).

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).

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} 

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.

1.10mins
Q10

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

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.

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}

3.54mins
Q13

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

2.07mins
Q14a

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

1.34mins
Q14b

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

1.40mins
Q14c

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

1.39mins
Q14d

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

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}.

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)

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.

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)

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.

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.

3.18mins
Q17

Write a brief description of each plane.

2x - 3y = 6

1.04mins
Q18a

Write a brief description of each plane.

x -3z = 6

0.48mins
Q18b

Write a brief description of each plane.

2y - z = 6

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)

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.

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}

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

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

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)

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

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

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.?

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.