Points defining plane - starting step?

Question

If the points $P, Q, R$, not all lying on the same straight line, have position vectors $a, b, c$ respectively, show that $(a \times b) + (b \times c) + (c \times a)$ is a vector perpendicular to the plane containing $P, Q, R$.

Answer 1

The vectors $\vec{RP}=(a-c)$ and $\vec{RQ}=(b-c)$ are independent vectors in the plane orientation. Therefore their cross product will be normal to the plane.

$\begin{align}(a-c)\times (b-c) &=a\times b -c\times b-a \times c + c\times c \\ &=a\times b +b\times c +c \times a + 0 \end{align}$