00:01
Let's start by finding the normal vector to the plane.
00:04
This has components equal to the coefficients in the equation for the plane.
00:10
So the x component will be negative 1, the y component will be 4, and the z component will be 4.
00:17
So we have negative i hat plus 4 j hat plus 4 k hat.
00:27
Now let's find the normal unit vector.
00:29
So we have n divided by its own magnitude to turn it into a unit vector.
00:37
So this is negative i hat plus 4 j hat plus 4 k hat divided by the square root of the sum of its components squared.
00:47
So that's negative 1 squared plus 4 squared plus 4 squared.
00:52
Okay so this is equal to, we have 1 over the square root of 1 plus 16 plus 16.
01:07
Okay 16 plus 16 is 32 plus 1 is 33.
01:11
So 1 over square root 33 times negative i hat plus 4 j hat plus 4 k hat.
01:20
Now let's pick a point q that is on the plane.
01:31
We'll just pick one that is convenient.
01:33
So i'm going to choose a point that has the same x and, sorry y and z components as the given point.
01:41
Okay so it will be 4 negative 4 and we'll use the equation for the plane to solve for the x component.
01:53
So we have negative 1 x plus 4 y so that's plus 4 times 4 plus 4 z so that's plus 4 times negative 4 is equal to negative 3.
02:10
Okay so that's negative x plus 16 minus 16 equals negative 3.
02:16
16 minus 16 is 0 so we just have negative x equals negative 3 so x equals positive 3.
02:25
Okay that's our x component...