Find the vector components of $\vec u$ in the direction of $\vec v$

Question

Find the vector components of $\vec u$ in the direction of $\vec v$

Answer 1

You have that $v=(1,2)$ and $u=(-1,-1)$. first you need to find the unitary vector in the same direction as $v$, we will call that $u_v$ and $u_v=\frac{v}{|v|}$. We know that $|v|=\sqrt 5$, so our unitary vector will be: $u_v=(\frac{1}{\sqrt 5},\frac{2}{\sqrt 5})$. Now to find the conponent of $u$ in the $v$ direction you simply calculate $u \cdot u_v$, giving you $(-1,-1)\cdot(\frac{1}{\sqrt 5},\frac{2}{\sqrt 5})= -(\frac{1}{\sqrt 5}+\frac{2}{\sqrt 5})= -\frac{3}{\sqrt 5}$