# Distance Between Point and Plane

#### USNAVY

Find the distance between the point and the plane.

Point: P(0, 0, 0)

Line: 2x + 3y + z = 12

My Work

Let n = normal vector

n = <2, 3, 1>

I now need a second point. Call it Point Q.

Let y = 0 and z = 0 in the given line.

2x + 3(0) + 0 = 12

2x + 0 = 12

2x = 12

x = 12/2

x = 6

Point Q is (6, 0, 0).

I also need vector PQ.

PQ = <6-0, 0-0, 0-0>

PQ = <6, 0, 0>

To find the distance, we use the distance formula below.

D = |PQ*n|/||n||

After plugging the information above into the formula, my answer is D = root{14}/7.

[6*root{14}]/7.

Why am I wrong?

#### HallsofIvy

MHF Helper
Find the distance between the point and the plane.

Point: P(0, 0, 0)

Line: 2x + 3y + z = 12

My Work

Let n = normal vector

n = <2, 3, 1>

I now need a second point. Call it Point Q.
No, you need to find the point at which this line intersects the given plane.

Let y = 0 and z = 0 in the given line.

2x + 3(0) + 0 = 12

2x + 0 = 12

2x = 12

x = 12/2

x = 6

Point Q is (6, 0, 0).

I also need vector PQ.

PQ = <6-0, 0-0, 0-0>

PQ = <6, 0, 0>

To find the distance, we use the distance formula below.

D = |PQ*n|/||n||

After plugging the information above into the formula, my answer is D = root{14}/7.
Okay, you have found the distance from P to this arbitrary point Q. What does that have to do with the distance from P to the plane?

[6*root{14}]/7.

Why am I wrong?
Why should you be right? You find an arbitrary second point on the line and determine the distance from (0, 0, 0) to that. That has nothing to do with the plane! You need to find the point at which the line through (0, 0, 0), perpendicular to the plane, intersects the plane!

Any line perpendicular to this plane has "direction" given by the vector <2, 3, 1>. Such a line, passing through (0, 0, 0), has parametric equation x= 2t, y= 3t, z= t.
To determine where that line intersects the plane, replace (x, y, z) in the equation of the plane and solve for t.

The point, Q, you found has nothing to do with the plane. It is just another point on the line. You could find a second, arbitrary point on the line in order to use the "two point" form for the line. But in three dimensions, that involves first finding the "direction vector" of the line- and that is the normal vector to the plane so you would be doing the same work two different ways.

#### Plato

MHF Helper
Find the distance between the point and the plane.
Point: P(0, 0, 0)
Line: 2x + 3y + z = 12
My Work
Let n = normal vector
n = <2, 3, 1>
PQ = <6, 0, 0>

To find the distance, we use the distance formula below.
D = |PQ*n|/||n|| YES!
my answer is D = root{14}/7.