Suppose you have 2 vectors $\vec a = (1,1,2)$ and $\vec b = (3,4,-2)$, how would you write $\vec a$ as the sum of 2 vectors $\vec c$ and $\vec d$ where $\vec c$ is in the direction of $\vec b$ and $\vec d$ is perpendicular to $\vec b$.
I think $\vec c$ may be the dot product of $\vec a$ and $\vec b$ multiplied by $\vec b$'s unit vector and $\vec d$ may be the cross-product of $\vec a$ and $\vec b$ as that is orthogonal to $\vec b$?
Thanks!
Best Answer
What you want looks like this $$ \vec a = \alpha \vec b + \vec d, \text{ where } \vec d\perp \vec b $$ Hence, $$ \langle \vec a,\vec b\rangle = \alpha \langle \vec b,\vec b\rangle $$ This helps you find $\alpha$. Now just take $\vec d := \vec a-\alpha \vec b$ and you are done!