8.5 The Cartesian Equation of a Plane
Chapter
Chapter 8
Section
8.5
Solutions 20 Videos

A plane is defined by the equation x -7y - 18z = 0.

a) What is a normal vector to this plane?

b) Explain how you know that this plane passes through the origin.

c) Write the coordinates of three points on this plane.

1.24mins
Q1

A plane is defined by the equation 2x - 5y = 0.

a) What is a normal vector to this plane?

b) Explain how you know that this plane passes through the origin.

c) Write the coordinates of three points on this plane.

1.16mins
Q2

A plane is defined by the equation x= 0.

a) What is a normal vector to this plane?

b) Explain how you know that this plane passes through the origin.

c) Write the coordinates of three points on this plane.

1.17mins
Q3

b) A plane has a normal of \vec{n} = (-\displaystyle{\frac{1}{2}}, \displaystyle{\frac{3}{4}}, \displaystyle{\frac{7}{16}}) and passes through the origin. Determine the Cartesian equation of this plane.

0.27mins
Q4b

A plane is determined by a normal, \vec{n} = (1, 7, 5), and contains the point P(-3, 3, 5). Determine a Cartesian equation.

0.00mins
Q5

Determine the Cartesian equation of the plane that contains the points A(-2, 3, 1), B(3, 4, 5), and C(1, 1, 0).

1.53mins
Q7

The line with vector equation \vec{r} = (2, 0, 1) + t(-4, 5, 5), s\in \mathbb{R}, lies on the plane \pi, as does the point P(1, 3, 0). Determine the Cartesian equation of \pi.

2.08mins
Q8

Determine unit vectors that are normal to each of the following planes:

3x - 4y + 12z - 1 = 0

0.56mins
Q9c

A plane contains the point A(2, ,2 -1) and the line \vec{r} = (1, 1, 5) + s(2, 1, 3), s \in \mathb{R}. Determine the Cartesian equation of this plane.

2.05mins
Q10

Determine the Cartesian equation of the plane containing the point (-1, 1, 0) and perpendicular to the line joining points (1, 2, 1) and (3, -2, 0).

2.01mins
Q11

a. Explain the process you would use to determine the angle formed between two intersecting planes.

b. Determine the angle between the planes x + 2y - 3z - 4 =0 and x + 2y - 1=0.

2.43mins
Q12

a) Determine the angle between the planes x + 2y - 3z - 4 = 0 and x + 2y - 1 = 0.

3.15mins
Q13a

Determine the Cartesian equation of the plane that passes through the point P(1, 2, 1) and is perpendicular to the line \displaystyle{\frac{x - 3}{-2}} = \displaystyle{\frac{y + 1}{3}} = \displaystyle{\frac{z +4}{1}}.

1.49mins
Q13b

What is the value of k that makes the planes 4x + ky - 2z + 1 = 0 and 2x + 4y - z + 4 = 0 parallel .

0.43mins
Q14a

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

What is the value of k that makes these two planes perpendicular?

0.32mins
Q14b

Can these two planes: 4x + ky - 2z + 1 = 0 and 2x + 4y - z + 4 = 0

ever be coincident? Explain.

0.35mins
Q14c

Determine the Cartesian equation of the plane that passes through the points (1, 4, 5) and (3, 2, 1) and it s perpendicular to the plane 2x - y + z - 1 = 0.

3.12mins
Q15

Determine an equation of the plane that is perpendicular to the plane x + 2y + 4= 0, contains the origin, and has a normal that makes an angle of 30^{\circ} with the z-axis.

Determine the equation of the plane that lies between the points (-1, 2, 4) and (3, 1, -4) and is equidistant from them.