Find two vectors $u$ and $v$ such that: $(i)$ both $u$ and $v$ are perpendicular to $w= (1,-1,-1)$ and $(ii)$ $u$ is perpendicular to $v$
So, I know how to find a vector, $u = (1,2,-1)$, that is perpendicular to $w$, but how do I find the third vector, $v$ that is perpendicular to both $u$ and $w$?
I've already tried $$v\cdot (u+w)=v\cdot u \; + v\cdot w \\
$$
But the result that I get does not allow for $u \cdot v = 0$ and $v \cdot w = 0$
What am I doing wrong? How should I approach solving something like this?
Please note that I'm taking Linear Algebra I, so keep the language/explanation to a level that matches my knowledge. Thank you.
Best Answer
HINT
Here is a sketch of the solution, which you may be more interested in the future.
Take any vector $x\in\mathbb{R}^{3}\backslash\{0,w\}$ then remove its projection on the direction of $w$.
The result of such operation is going to be denoted by $u$, which is orthogonal to $w$.
Precisely, one has that \begin{align*} u = x - \frac{\langle x,w\rangle}{\|w\|^{2}}w \end{align*}
Once you have $u$ and $w$, you can take the cross product $v = u\times w$, and you are done.
Hopefully this helps !