## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$v \cdot w=(i+j) \cdot (-i+j-k) \\= (1)(-1) +(1)(1)+(0)(-1) \\=0$ Therefore, the angle between the two vectors $v$ and $w$ is $90^{\circ}$.
We know that when the dot product between the two vectors is $0$, then they are perpendicular or orthogonal. This means that the angle between the two vectors is $90^{\circ}$. Let us consider two vectors $v=pi+qj+rk$ and $w=xi+yj+zk$. Then we have: $v \cdot w=(pi+qj+rk) \cdot xi+yj+zk =px+qy+rz$ In our case: $v \cdot w=(i+j) \cdot (-i+j-k) \\= (1)(-1) +(1)(1)+(0)(-1) \\=0$ Therefore, the angle between the two vectors $v$ and $w$ is $90^{\circ}$.