Answer
(a) $\lt 13,-14,5 \gt$
(b) $\frac{ \sqrt {390}}{2}$
Work Step by Step
(a) Given: $P(0,-2,0),Q(4,1,-2)$ and $R(5,3,1)$
$PQ ^\to=\lt4-0,1-(-2),-2-0\gt=\lt 4,3,-2 \gt$
$PR ^\to=\lt 5-0,3-(-2), 1-0\gt=\lt 5,5,1 \gt$
$\lt 4,3,-2 \gt \times \lt 5,5,1 \gt=\lt 13,-14,5 \gt$
(b) Area of a vector with vertices at P,Q, and R is
$ Area=\frac{1}{2}|PQ ^\to \times PR ^ \to|$
$PQ ^\to \times PR ^ \to=\lt 4,3,-2 \gt \times \lt 5,5,1 \gt=\lt 13,-14,5 \gt$
$|PQ ^\to \times PR ^ \to|=\sqrt {(13)^2+(-14)^2+(5)^2}= \sqrt {390}$
$ Area=\frac{1}{2}|PQ ^\to \times PR ^ \to|=\frac{ \sqrt {390}}{2}$
