Hi, I'm struggling with this problem:

For $\displaystyle \vec{u} = $[-4, 1, 10]^T and $\displaystyle \vec{v} =$ [−12, −6, 8]^T find the vectors $\displaystyle \vec{u1}$ and $\displaystyle \vec{u2}$ such that:

(i) $\displaystyle \vec{u1}$ is parallel to $\displaystyle \vec{v}$

(ii) $\displaystyle \vec{u2}$ is orthogonal to $\displaystyle \vec{v}$

(iii) $\displaystyle \vec{u} = \vec{u1} + \vec{u2}$

I figured I should firstly try and find u2 by (ii), and then after I found that I would be able to use (iii) to get u1. This approach didn't work out too well for me, heh. Basically I tried to set it up with dot product, and solve for u2:

$\displaystyle \vec{v} * \vec{u2} = 0 $

Didn't work out, just ended up with something like: -12a - 6b + 8c = 0

Where a, b, c are the numbers in u2.

From there I couldn't see what more I could do to find a, b, c. Basically, I don't really know how to approach this problem

.

Anyone mind helping out a math newbie? Thanks in advance!