Answer
(a) $ Q=(5,3,2)$.
(b) the unit vector in the direction of $\overrightarrow{ PQ}$ is given by $$ \left\langle \frac{3}{\sqrt{35}},-\frac{1}{\sqrt{35}},\frac{5}{\sqrt{35}}\right \rangle.$$
Work Step by Step
(a) Let $ Q=(a,b,c) $; then we have $$\overrightarrow{ PQ}=Q-P=(a,b,c)- (1,4,-3)=\langle a-1,b-4,c-3 \rangle $$ now since $\langle a-1,b-4,c-3 \rangle $ and $\langle 3,-1,5 \rangle $ are equivalent, then we have $\langle a-1,b-4,c-3 \rangle=\langle 3,-1,5 \rangle $ and hence
$$ a-1=4, \quad b-4=-1, \quad c+3=5.$$ That is, $ Q=(5,3,2)$.
(b) The unit vector in the direction of $\overrightarrow{ PQ}$ is given by $$\frac{\langle 3,-1,5 \rangle}{\sqrt{9+1+25}}=\left\langle \frac{3}{\sqrt{35}},-\frac{1}{\sqrt{35}},\frac{5}{\sqrt{35}}\right \rangle.$$
