Answer
(a) $ b=-\frac{1}{2}$.
(b) $ b=0 $ or $ b=\frac{1}{2}$.
Work Step by Step
Since the vectors are orthogonal, then we have
(a) $$\langle b,3,2\rangle \cdot \langle1,b,1\rangle=0\Longrightarrow b+3b+2=0.$$
$$4b=-2$$
Hence $ b=-\frac{1}{2}$.
(b) $$\langle 4,-2,7\rangle \cdot \langle b^2,b,0\rangle=0\Longrightarrow 4b^2-2b=0.$$
$$4b(b-\dfrac{1}{2})=0$$
Hence $ b=0 $ or $ b=\frac{1}{2}$.
