Question

Write the vector u=(2,3,-1) as the sum of two vectors, one parallel to v=(1,0,-3) and the other orthogonal to v=(1,0,-3)

Answer 1

Parallel vectors are scalar multiples of each other
the dot product orthogonal vectors is zero.

Answer 2

I know this, I can't figure out the system of equation. Let's say I have vector w for orthogonal and z for parallel i have this:

w1*1+w2*0+w3*-3=0
k(z1,z2,z3)=v

Answer 3

then What do i do? equate all the x values to the x value of u? it doesn't work, system becomes inconsistent. However I know the problem has a unique solution.