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

$b=-1$
Let us consider two vectors $v=pi+qj$ and $w=xi+yj$ The dot product is given as $v \cdot w=(pi+qj) \cdot (xi+yj) =px+qy$ We know that when the dot product between two vectors is $0$, then they are perpendicular or orthogonal. This means that the angle between the two vectors is $90^{\circ}$. We have the given vectors : $v=i +j$ and $w=i+bj$ Since, both vectors are orthogonal, then $v \cdot w=0$ $v \cdot w=(i+j) \cdot (i+bj) =0 \implies (1)(1) +(1)(b) \\=0 \\ \implies 1+b =0 \\ \implies b=-1$