This is the question I am trying to solve: My cat is standing on the plane $3x+2y+z = 4.$ Its head is at $(0,-1,-3)$ and the tip of its tail is at $(-4,-2,-2).$ Measured from the plane, what is the difference between the height of my cat's head and my cat's tail?
I originally just thought the answer was just the distance between the two points, and I got ${3 \sqrt{2}}{},$ however, it is wrong. I don't know what else to do. Any ideas?
Best Answer
The twist here is that the cat is not standing perpendicular to the plane and is along the vector $(4, 1,-1) $. So, you have to project the length $3\sqrt 2$ along $(4, 1,-1) $ on the normal vector $(3, 2,1) $.