Let's start with a simpler problem. If I have two vectors V = (v1, v2, v3, v4) and X = (x1, x2, x3, x4), then what is the orthogonal projection of V onto X?
Orthogonal projection is essentially taking the "component" of V along the "direction" of X. For example, if I am moving in 3 dimensions and have a velocity V = (1, 2, 3) and I want to know how fast I am moving in the z-direction, then I would compute the component of velocity in the direction of z which is (0,0,3). In general, we compute the orthogonal projection of V onto W using the dot product. The formula for it is (V · W/|W|^2) W.
Where does this formula come from? While we won't derive it here, we can understand that since V · W = |V| |W| cos (θ) where θ is the angle between the vectors, then (V · W)/|W| = |V| cos (θ) is the magnetude of the component of V in the direction of W. The other part, W/|W|, is the unit vector in the W direction.
So if we understand how to calculate the component of of a vector in the direction of another vector, how then do we calculate the orthogonal projection of a vector onto a plan spanned by two vectors? Well, if we can find an orthogonal basis for that plane and compute the components of our vector V on each of the basis vectors, then we can just sum the results to get the project of V onto the plane. If we look closely at X1 and X2, it turns out they are already orthogonal! That is, X1 · X2 = (-3)*(-4) + 4*(-5) + 2*4 + (-4)*0 = 12 - 20 + 8 = 0. Therefore, if we compute the projection of V onto X1 and onto X2, respectively, then the sum of these two projections will be the projection of V onto the plane spanned by X1 and X2.
First, the projection of V onto X1 is (V · X1)/|X1|^2 X1 = (16*(-3)+12*4+18*2+14*(-4))/((-3)^2 +4^2+2^2 + (-4)^2) (-3,4,2,-4) = (-20)/(45) (-3,4,2,-4) = -(4/9) (-3,4,2,-4).
The projection of V onto X2 is (V · X2)/|X2|^2 X2 = (16*(-4)+12*(-5)+18*4+14*0)/((-4)^2 +(-5)^2+4^2 + 0^2) (-4,-5,4,0) = (-52)/(57) (-4,-5,4,0)
Now adding these two vectors together will give us our final answer: (4/3 + 208/57,-16/9+260/57,8/9-208/57,0) = (284/57,476/171,-472/171,0).
