how to find a vector?

Question

Consider the inner product

$\left<(a_1, a_2),(b_1, b_2)\right> = 2a_1b_1 − a_1b_2 − a_2b_1 + 5a_2b_2$ on $R^2$

Write down a vector which is orthogonal to$ (1, 0)$ and has norm $1.$

My solution : i convert them into matrix

$A= \begin{bmatrix} 2&-1\\-1&5\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix}=\begin{bmatrix}2\\-1\end{bmatrix}$

Now i don't know how to find a vectors which is orthogonal to$ (1, 0)$ and has norm $1.$

Answer 1

You can simply write two equations for both conditions and then solve the system. They would be $2b_1-b_2=0$ and $2b_1^2-2b_1b_2+5b_2^2=1$.