# Calculating the projection of a point onto a plane

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Milton Niklasson on 20 Sep 2021
Edited: Yongjian Feng on 20 Sep 2021
I need to calculate the orthogonal projection of the point x=(3.5,1.5,−1.5) on the plane 4x−4y+4z=12 and also calculate the reflection of point x in the same plane. I have tried my best but i cannot figure out how to do this using MATLAB, any suggestions?
John D'Errico on 20 Sep 2021
This is surely homework. You telling us that you have tried your best merely says that you gave up. And since it is homework, you need to be the one doing the thinking.
Surely you can find the normal vector to the plane defined by that equation? What does it tell you?

Yongjian Feng on 20 Sep 2021
Edited: Yongjian Feng on 20 Sep 2021
Orthogonal projection can be thought as the (smallest) distance from this point to the plane.
Assume that the projection is (a,b,c). Then you have distance d as:
d^2 = (a-3.5)^2+(b-1.5)^2+(c+1.5)^2
Because (a,b,c) is a point on the plane, so you also have
4*a-4*b+4c = 12
Then you can combine the above two, and get
d^2 = (a-3.5)^2+(b-1.5)^2+(b-a+4.5)^2
Now you need to minimize d basically. This will give you a, b, and c.