see this thread
Finding decomposition of vectors
Take n = 2i + j - k as a normal vector of the plane p. Decompose the vector QO into the sum of two vectors; one of them is parallel to n and the other one is orthogonal to n.
So far I got, Q(0,0,-1) and (0,0,0) for the origin. and QO = (0,0,1).
n=2i+j-k
o=ai+bj+ck
(2+a)i+(1+b)j+(-1+c)k=0i+0j+k
I get a=-2, b=-1, c=2. Therefore o = -2i-j+2k
But the dot product of these two vectors is not equal to zero so the equation for o is incorrect. When i test i-j+k the dot product is zero but how would I know for sure without trial and error.
see this thread
Finding decomposition of vectors