I have the point P,Q and R given. I calculate the displacement vectors PQ and PR. If I then compute their cross product I get a vector orthogonal to the plane they're in. But the value of the cross product vector is not orthogonal to PQ and PR (it is not 0). Shouldn't it be 0?
Thank you
Best Answer
The Cross Product $v \times w$ is always orthogonal to both $v$ and $w$. This is easy to see by a direct calculation:
Write $v = \langle v_1, v_2 , v_3\rangle$ and $w = \langle w_1, w_2, w_3\rangle $. Then
$$ v \times w = \langle v_2 w_3 - v_3 w_2, v_3w_1 - v_1w_3, v_1w_2 - v_2 w_1 \rangle $$
which implies
$$\begin{align} v \cdot (v \times w) &= v_1 (v_2 w_3 - v_3 w_2) + v_2 (v_3 w_1 - v_1 w_3) + v_3 (v_1 w_2 - v_2 w_1) \\ &= v_1 v_2 w_3 - v_1 v_3 w_2 + v_2 v_3 w_1 - v_1 v_2 w_3 + v_1 v_3 w_2 - v_2 v_3 w_1 = 0. \end{align}$$ The same calculation works to show $w \cdot (v \times w) = 0$.