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## Homework Statement

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)

## Homework Equations

orthogonal vector would imply that v"dot"w = 0 ( dot product)

Parallel vector would imply that v=kz

OR the crossproduct of both is 0

## The Attempt at a Solution

My attempt went as follow:

I defined 2 vectors that w+z = u where u=(2,3,-1)

Vector w is orthogonal to v and therefore --> w"dot"v = 0 --> w

_{1}*1+w

_{2}*0+w

_{3}*-3=0

And that z= kv which would give z= (1*k,0*k,3*k)

I thought that I could define any vector orthogonal that would give 0, for instance (3,1,1) would give 0.

However the system always ends up being inconsistent, I don't know where I might be going wrong :/