Answer
$$\theta=90^{\circ}$$
Work Step by Step
Given $$v=(0,-2,1,4,1),w=(-3,1,-1,0,3)$$
Let $\theta$ be an angle between the vectors $v$ and $w .$
We know that
$$\cos \theta=\frac{\langle v,w\rangle}{\|v\|\|w\|}$$ .
Since we have
$\langle v, w\rangle= 0 \cdot(-3)+(-2) \cdot 1+1 \cdot(-1)+4 \cdot 0+1 \cdot 3=0$
So we get:
$$\cos \theta=\frac{\langle v,w\rangle}{\|v\|\|w\|}=\frac{0}{\|v\|\|w\|}=0$$
The angle between $v$ and $w$ is $90^{\circ},$ ie. $v$ and $w$ are perpendicular.