Write vector $\vec{u}$ as a sum of vector parallel to $\vec{v}$ and perpendicular to $\vec{v}$
$\vec{u}$ = $< 2,4,2 >$ and $\vec{v}$ = $< 1,2,-1 >$
I was pretty sure that the way to solve it was to find the parallel vector by doing
$
u_1
= \left(\frac{|u|\cdot |v|}{|v| \cdot |v|}\right)v $
and then subtract that vector from $\vec{u}$ to get the perpendicular one.
That left me with $\vec{u}$ = $< 2,4,-2 > + < 0,0,4 >$ which can't be correct because the second vector isn't perpendicular to $\vec{v}$.
What am I doing wrong, or did I mess up my algebra?
Best Answer
The standard way is\begin{align}\vec u&=\left\langle\vec u,\frac{\vec v}{\left\|\vec v\right\|}\right\rangle\frac{\vec v}{\left\|\vec v\right\|}+\left(\vec u-\left\langle\vec u,\frac{\vec v}{\left\|\vec v\right\|}\right\rangle\frac{\vec v}{\left\|\vec v\right\|}\right)\\&=\left(\frac{4}{3},\frac{8}{3},-\frac{4}{3}\right)+\left(\frac{2}{3},\frac{4}{3},\frac{10}{3}\right).\end{align}