Find the vector by the following criteria

Question

I want to find the vector that meets the following: $$X\parallel (2,1,-1)$$ $$X*(2,1,-1)=3$$ what I did so far is :
$$2x+y+z=3$$ I know that parallel vectors the angle is $180$ or $0$.
how to continue from here?

Note: X is vector.
Thanks!

Answer 1

The easiest way is to notice that $X\parallel Y$ iff $X = a Y$ for some $a\in \Bbb R$. That is, you know that $$ X = (2a,a,-a) $$ and that $2a\cdot 2 + a\cdot 1+(-a)\cdot (-1) = 3$ which will give you $a$.

An "alternative" way which is not really an alternative is so mention that you have two conditions: $X\parallel Y$ and $X\cdot Y = 3$. The first is equivalent to $X = aY$ as we already discussed, so the second now turns to be $$ 3 = X\cdot Y = aY\cdot Y = a\|Y\|^2 \iff a = \frac3{\|Y\|^2} $$ where $\|Y\|$ is the length of $Y$, which you can find using squares and square roots.