I have been struggling with a homework question and I would like to get some help.
The first section of the question it prove the Triple Product Identity: $ (\vec u \times \vec v)\times \vec w= (\vec w \cdot \vec u)\vec v - (\vec v \cdot \vec w)\vec u $
Which I proved.
The second section says :
"let $ \vec u , \vec v \neq \vec 0 $ and $\alpha \in \mathbb{R} $
Is vector $ \vec w $ exists, such that $ \vec u \times \vec w = \vec v$ and $\vec u \cdot \vec w =\alpha $ ?
In cases where $\vec w$ exists, give example for such $\vec w$.
In cases where $\vec w$ does not exist, explain why."
So in order to find such $\vec w$, $\vec u$ and $\vec v$ must be perpendicular. If $\alpha = 0$ I managed to find such $\vec w$. But when $\alpha \neq 0$ I didn't manage to find such $\vec w$ nor prove that it doesn't exist. I tried to use $ \vec u \times \vec w = \vec v$ and then write $ (\vec u \times \vec w )\times \vec v= \vec v \times \vec v =\vec 0 $ and then use the first section, but it didn't lead me to success.
Thank you for your time.
If such a $\vec w$ exists, then
$$\vec u\times \vec v=\vec u\times(\vec u\times \vec w)=-(\vec u\times \vec w)\times \vec u=-[(\vec u\cdot \vec u)\vec w-(\vec u\cdot \vec w)\vec u]=\alpha \vec u-\vec u^2\vec w$$
from which you can find $\vec w$:
$$\vec w=\frac{1}{\vec u^2}\left(\alpha \vec u-\vec u\times\vec v\right)$$
Then:
$$\vec u\cdot \vec w=\alpha \frac{\vec u^2}{\vec u^2}=\alpha$$
And
$$\vec u\times\vec w=\frac{1}{\vec u^2}(\vec u \times \vec v)\times \vec u=\frac{1}{\vec u^2}\left[(\vec u^2)\vec v-(\vec u\cdot \vec v)\vec u\right]=\vec v-\frac{\vec u\cdot \vec v}{\vec u^2}\vec u$$
Therefore, $\vec u\times \vec w=\vec v$ iff $\vec u\cdot \vec v=0$.