So I keep self learning linear algebra but problems keep arising. I'm trying to solve this particular one:
Problem
Find all the vectors $v$ on $\mathbb{R}^3$ wich are orthogonal to the vector $s = [1,2,-1]$ and have norm (magnitude) $= 2$.
I try to solve the problem but I'm not sure how to express the solution, this is my solution:
Solution
I model $v$ as $[v_1,v_2,v_3]$ and given that $v$ is orthogonal to $s$ then the dot product of them will be $0$ and so:
$v_1+2v_2-v_3 = 0$; $v_1 = v_3-2v_2$; $v_2 = (1/2)(v_3-v_1)$ and $v_3 = v_1+2v_2$
But that only solves the first part of the problem. For the second I now the norm is 2 so:
$\|v\| = 2 = \sqrt{v_1^2 + v_2^2 + v_3^2} $
I'm stuck here, I don't know where to follow or if just defining that both conditions must be satisfied is enough. Thanks in advance for any guidance.