How to find the vector of length $5$?

Question

Given vector $\mathbf a = -2\hat i + \hat j + \hat k$ and $\mathbf b = \hat i - 3\hat j + \hat k.$
Find the vector of length $5$ perpendicular to the plane containing vector $\mathbf a$ and $\mathbf b$. I have worked out the perpendicular of $\mathbf a$ and $\mathbf b$, the vector is $4\hat i + 3\hat j + 5\hat k$. But I do not know how to find the vector with length $5$.

Answer 1

If you have a vector $v$, you can find a vector $w$ of length $\ell$ in the same direction as $v$ by writing $w=\ell v/\|v\|$. The quantity $\|v\|$ is the length of $v$ (in Euclidean space you can compute this using the Pythagorean Theorem), so $v/\|v\|$ has length $1$, so $\ell v/\|v\|$ has length $\ell$.