Yet another "Help me with my math" thread

[Social Bacon]

So I need to find the distance from a point to a plane in 3-dimensional space.
Where the plane is represented by the scalar equation: 2x + y - 2z + 9 = 0
And the point is (2, 3, -1). We shall call it point A.

Knowing that the shortest distance between the plane and point will be along the line that is perpendicular to the plane we can use the plane's normal vector (which is n(2, 1, -2)).

We shall call the point of the plane closest to A "Q".

Therefore the line that passes through the segment AQ can be represented by the parametric equations:
x = 2 + 2k
y = 3 + k
z = -1 - 2k

Subbing the equation of line AQ into the equation of the place will give us the value of k that puts the line on point Q.

2(2 + 2k) + (3 + k) - 2(-1 - 2k) + 9 = 0
4 + 3 + 2 + 9 + 4k + k + 4k = 0
18 + 9k = 0
k = -2

Now we can sub this k value back into the parametric equation of the line to find point Q.
x = 2 + 2(-2)
y = 3 - 2
z = -1 - 2(-2)
Therefore Q(-2, 1, 3)

Therefore line segment AQ(-4, -2, 4).
And thus the magnitude of AQ will be the distance between point A and the plane.
|AQ| = sqrt (((-4)^2) + ((-2)^2) + (4^2))
|AQ| = sqrt (16 + 4 + 16)
|AQ| = sqrt (36)
|AQ| = 6

Therefore the distance from the point to the plane is 6 units.

Is my theory all correct? I made this all up on the spot without help of a teacher or textbook, so I have no idea if I made some fundamental error.

 

[Druckles]

Didn't do 3D planes *stretches arms and cracks knuckles* so I can't help you.

If you'd asked me about the shortest distance to a 3D line, I would've been fine

Considering the lack of input from the last Maths thread, and considering this is much more advanced, I don't think you're going to get much of a response.

 

[99.vikram]

1. Substitute (2,3,-1) in place of (x,y,z) to find the value of (2x + y - 2z + 9). This works out to 18.

2. Find rt(sq(2) + sq(1) + sq(-2)). [rt: root of, sq: square of]

3. Divide result 1 by result 2 to get 6.

It's a much simpler method which can be derived by either vectors or 3D geometry. (Do you need the derivation?)

 

[Social Bacon]

As a matter of personal principle I refuse to use a method that I do not fully understand. So as long as you can confirm that my method is correct I'm pleased

I'm assuming that I could get the derivation if I just went through my method with the variables.

 

[Sheepo]

You're on the right track, just divide by zero.