Find $\|\mathbf{v}\|$ and $\|\mathbf{v+u}\|$ knowing $\|\mathbf{u}\|=3$; $\angle (\mathbf{u,v})=\frac{\pi}{4}$;$\angle (\mathbf{u+v,u})=\frac{\pi}{6}$

Question

I need to find $\|\mathbf{v}\|$ and $\|\mathbf{v+u}\|$. The data I have is the following:

$\|\mathbf{u}\|=3$

$\angle (\mathbf{u,v})=\frac{\pi}{4}$

$\angle (\mathbf{u+v,u})=\frac{\pi}{6}$

I have tried to use the equation of $\cos \theta=\frac{\mathbf{u\cdot v}}{\|\mathbf{u}\|\cdot\|\mathbf{v}\|}$ but I don't get anywhere. Specifically, I get 2 equations with $u\cdot\ v$, $\|\mathbf{v}\|$ and $\|\mathbf{v+u}\|$ and I don't know where to go. Any hint or help will be appreciated, thank you.

Answer 1

"any hint..."

The Law of Sines: We know one side of a triangle, and the angles at the two ends of that side, find the other two sides. So: either prove this using the LoS, or consider this a request of how to prove the LoS.

Answer 2

You can find a third equation by expanding $(\bf u+\bf v)\cdot(\bf u+\bf v)$: $$ \| \bf u+\bf v\|^2=(\bf u+\bf v)\cdot(\bf u+\bf v) =(\bf u+\bf v)\cdot \bf u + \bf u\cdot \bf v + \|\bf v\|^2 $$ and substituting what you know and what you've already calculated.