9.6 The Distance from a Point to a Plane
Chapter
Chapter 9
Section
9.6
Lectures 3 Videos  Introduction to Shortest distance between Point and Line in 3D Finding Distance between two Parallel Planes Shortest Distance between Two Skew Lines
Solutions 16 Videos

A student is calculating the distance d between point A(-3,2,1) and the plane with equation 2x+y+2z+2=0. The student obtains the following answer:

d=\displaystyle{\frac{|2(-3)+2+2(1)+2|}{\sqrt{2^2+1^2+2^2}}}=\displaystyle{\frac{0}{3}}=0

a) Has the student done the calculation correctly? Explain.

b) What is the significance of the answer 0? Explain.

0.58mins
Q1

Determine the

distance from A(3, 1, 0) to the plane with equation 20x -4y + 5z + 7 = 0.

0.48mins
Q2a

Determine

the distance from B(0, -1, 0) to the plane with equation 2x + y + 2z - 8 = 0.

0.52mins
Q2b

Determine

the distance from B(5, 1, 4) to the plane with equation 3x -4y - 1 = 0.

0.51mins
Q2c

Determine

the distance from D(1, 0, 0) to the plane with equation 5x -12y = 0.

0.40mins
Q2d

Determine

the distance from E(-1, 0, 1) to the plane with equation 18x - 9y + 18z - 11 = 0.

0.52mins
Q2e

For the planes \pi_1:3x + 4y - 12z - 26 = 0 and \pi_2: 3x + 4y -12z + 39 = 0, determine

• the distance between \pi_1 and \pi_2.
2.13mins
Q3a

For the planes \pi_1:3x + 4y - 12z - 26 = 0 and \pi_2: 3x + 4y -12z + 39 = 0, determine

i) the equation for a plane midway between \pi_1 and \pi_2.

ii) the coordinates of a point that is equidistant from \pi_1 and \pi_2.

2.01mins
Q3bc

Determine the following distances:

• the distance from P(1, 1, -3) to the plan with equation y + 3 = 0.
0.43mins
Q4a

Determine the following distances:

b) the distance from Q(-1, 1, 4) to the plan with equation x- 3 = 0.

0.39mins
Q4b

Determine

the distance from R(1, 0, 1) to the plan with equation z + 1 = 0.

0.20mins
Q4c

Points A(1, 2, 3), B(-3, -1, 2), and C(13, 4, -1) lie on the same plane. Determine the distance from P(1, -1, 1) to the plane containing these three points.

2.49mins
Q5

The distance from R(3, -3, 1) to the plane with equation Ax - 2y + 6z = 0 is 3. Determine all possible value(s) of A for which this is true.

1.25mins
Q6a

Determine the distance between the lines \vec{r} = (0, 1, -1) + s(3, 0, 1), s \in \mathbb{R} and \vec{r} = (0, 0, 1) + t(1, 1, 0) t\in \mathbb{R}.

2.38mins
Q7

Calculate the distance between the lines L_1: \vec{r} = (1, -2, 5) + s(0, 1, -1), s \in \mathbb{R} and  L_2: \vec{r} = (1, -1, -2) + t(1, 0, -1), t \in \mathbb{R}.

For L_1: \vec{r} = (1, -2, 5) + s(0, 1, -1), s \in \mathbb{R} and  L_2: \vec{r} = (1, -1, -2) + t(1, 0, -1), t \in \mathbb{R}.
• Determine the coordinates of points on these lines that produce the minimal distance between L_1 and L_2.