Find a vector orthogonal to both $u $and $j+k$

Question

So, $u=i-3j+2k$

I understand how to solve these type of problems with two given vectors, however I am lost on this.

How do I find a vector orthogonal to both $u$ and $j+k$?

Answer 1

Let $v = (a,b,c)$ such that $v \perp u$, and $v \perp j+k = (0,1,1) \Rightarrow a - 3b + 2c = 0 = b + c \Rightarrow a = -5c, b = -c \Rightarrow v = (-5c,-c,c) = c(-5,-1,1), c \in \mathbb{R}$.

Answer 2

If $u=\langle 1,-3,2\rangle$ and $v=\langle 0,1,1\rangle$, then $u \times v=\langle-5,-1,1\rangle$ is orthogonal to both u and v.