Answer
(a) $\lt -4,7,-10 \gt$
(b) $\frac{ \sqrt {165}}{2}$
Work Step by Step
(a) Given: $P(-1,3,1),Q(0,5,2)$ and $R(4,3,-1)$
$PQ ^\to=\lt 0-(-1),5-3,2-1\gt=\lt 1,2,1 \gt$
$PR ^\to=\lt 4-(-1),3-3, -1-1\gt=\lt 5,0,-2 \gt$
$\lt 1,2,1 \gt \times \lt 5,0,-2 \gt=\lt -4,7,-10 \gt$
(b) Area of a vector with vertices at P,Q, and R is
$ Area=\frac{1}{2}|PQ ^\to \times PR ^ \to|$
$PQ ^\to \times PR ^ \to=\lt 1,2,1 \gt \times \lt 5,0,-2 \gt=\lt -4,7,-10 \gt$
$|PQ ^\to \times PR ^ \to|=\sqrt {(-4)^2+(7)^2+(-10)^2}= \sqrt {165}$
$ Area=\frac{1}{2}|PQ ^\to \times PR ^ \to|=\frac{ \sqrt {165}}{2}$
