## Differential Equations and Linear Algebra (4th Edition)

$$\theta=90^{\circ}$$
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.