How to find two orthogonal vectors $a$ and $b$ such that $a$ is the projection of vector U on V and $a+b=U$

Question

Given that U=<2,6> and V=<9,2> find two orthogonal vectors $a$ and $b$ such that $a$ is the projection of vector U on V and $a+b=U$

I first wanted to know how to even do this problem and understand it.

• I don't understand the "orthogonal vectors" part, I was wondering if this was referring to the legs the are formed when you project two vectors?

• If not how does one get two orthogonal vectors out of this problem?

• And also assuming that it is the legs, then I must do $Proj_{v}{u}$ right? And if I do that then I only get one vector, then what do I do? And also if this is the case, then is the fact that $a+b=U $ now useless?

Answer 1

• $a$ and $b$ orthogonal means that $\langle a, b \rangle =0$
• $a$ is the projection of $U$ on $V$ gives that $a=\lambda V$

Then $\langle b, V \rangle=0$ and $\langle U, V \rangle=\langle a, V \rangle$. Then $\lambda \Vert V \Vert ^2 = \langle U, V \rangle = 30$. This gives $\lambda = \frac{6}{17}$.

Then $a=(\frac{54}{17}, \frac{12}{17})$ and $b=U-a=(\frac{-20}{17},\frac{90}{17})$