Which is the angle of the two vectors?

Question

We have the vectors $v=i+j+2k=(1,1,2)$ and $u=-i-k=(-1,0,-1)$.

I want to calculate the angle between $u$ and $v$ in radians using the cross product.

I have done the following:

\begin{align*}|v\times u|=|v|\cdot |u|\cdot \sin \theta &\Rightarrow \sin \theta=\frac{|v\times u|}{|v|\cdot |u|}=\frac{\sqrt{3}}{\sqrt{6}\cdot \sqrt{2}}=\frac{\sqrt{3}}{\sqrt{3}\cdot \sqrt{2}\cdot \sqrt{2}}=\frac{1}{2} \\ & \Rightarrow \theta=2\pi n+\frac{\pi}{6} \ \text{ or } \ \theta=2\pi n+\frac{5\pi}{6}, \ n\in \mathbb{Z}\end{align*}

Is everything correct?

Which of the values do we choose? Or are both valid?

Answer 1

The answer is $\frac{5\pi}{6}$ because$$\cos\theta=\frac{-1-2}{\sqrt6\cdot\sqrt2}=-\frac{\sqrt3}{2}$$