Answer
(a) $\lt -1,-7,6 \gt$
(b) $\frac{ \sqrt {86}}{2}$
Work Step by Step
(a) Given: $P(0,0,-3),Q(4,2,0)$ and $R(3,3,1)$
$PQ ^\to=\lt4-0,2-0,0-(-3)\gt=\lt 4,2,3 \gt$
$PR ^\to=\lt 3-0,3-0,1-(-3)\gt=\lt 3,3,4 \gt$
$\lt 4,2,3 \gt \times \lt 3,3,4 \gt=\lt 2(4)-3(3),3(3)-4(4),4(3)2-2(3)\gt =\lt -1,-7,6 \gt$
(b) Area of a vector with vertices at P,Q, and R is
$ Area=\frac{1}{2}|PQ ^\to \times PR ^ \to|$
$PQ ^\to \times PR ^ \to=\lt 4,2,3 \gt \times \lt 3,3,4 \gt=\lt 2(4)-3(3),3(3)-4(4),4(3)2-2(3)\gt =\lt -1,-7,6 \gt$
$|PQ ^\to \times PR ^ \to|=\sqrt {(-1)^2+(-7)^2+(6)^2}=\sqrt {86}$
$ Area=\frac{1}{2}|PQ ^\to \times PR ^ \to|=\frac{ \sqrt {86}}{2}$
