9.6 The Distance from a Point to a Plane
Chapter
Chapter 9
Section
9.6
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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.

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0.58mins
Q1

Determine the

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

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0.48mins
Q2a

Determine

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

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0.52mins
Q2b

Determine

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

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0.51mins
Q2c

Determine

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

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0.40mins
Q2d

Determine

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

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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.
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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.

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2.01mins
Q3bc

Determine the following distances:

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

Determine the following distances:

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

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0.39mins
Q4b

Determine

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

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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.

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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.

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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}.

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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}.

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2.27mins
Q8a

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.
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5.53mins
Q8b