Answer to Question #107895 in Analytic Geometry for Caylin

Question

Suppose u, v and w are vectors in 3-space,where u =(u1u2, u3), v =(v1, v2, v3) and w= (w1, w2, w3).

Express ( u * v) * w as a determinant.

Answer 1

$u=(u_1,u_2,u_3), v=(v_1,v_2,v_3), w=(w_1,w_2,w_3)$

$u\times v=\begin{vmatrix} \vec{i} &\vec{j}&\vec{k} \\ u_1 &u_2&u_3\\ v_1&v_2&v_3 \end{vmatrix}=\\ =\vec{i}\cdot(u_2v_3-u_3v_2)-\vec{j}\cdot(u_1v_3-u_3v_1)+\\ +\vec{k}\cdot(u_1v_2-u_2v_1)=\\ =(u_2v_3-u_3v_2,-u_1v_3+u_3v_1,u_1v_2-u_2v_1)\\ (u\times v)\times w=\\ =\begin{vmatrix} \vec{i} &\vec{j}&\vec{k} \\ u_2v_3-u_3v_2&-u_1v_3+u_3v_1&u_1v_2-u_2v_1\\ w_1&w_2&w_3 \end{vmatrix}$