Question

Suppose u and v are vectors in 3–space where u = (u1, u2, u3) and v = (v1, v2, v3). Evaluate u×v×u and v × u × u.

Answer 1

Submit $u=(u_1,u_2,u_3)$, $v=(v_1,v_2,v_3)$.

Calculate vector product of $u$ and $v$:

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

And obtain:

$u\times v\times u=\begin{vmatrix} \vec i & \vec j & \vec k \\ u_2v_3-u_3v_2 & u_1v_3-u_3v_1 & u_1v_2-u_2v_1 \\ u_1 & u_2 & u_3 \end{vmatrix} =\vec i(u_1u_3v_3-u_3^2v_1-\\ {}-u_1u_2v_2+u_2^2v_1)+\vec j(u_2u_3v_3-u_3^2v_2-u_1^2v_2+u_1u_2v_1)+\vec k(u_2^2v_3-u_2u_3v_2-\\ {}-u_1^2v_3+u_1u_3v_1)=(u_1u_3v_3-u_3^2v_1-u_1u_2v_2+u_2^2v_1,\,u_2u_3v_3-u_3^2v_2-\\ {}-u_1^2v_2+u_1u_2v_1,\,u_2^2v_3-u_2u_3v_2-u_1^2v_3+u_1u_3v_1).$

According to vector product properties $u\times v\times u=u\times (v\times u)=-(v\times u)\times u$, so

$v\times u\times u=-u\times v\times u=-(u_1u_3v_3-u_3^2v_1-u_1u_2v_2+u_2^2v_1,\,u_2u_3v_3-u_3^2v_2-\\ {}-u_1^2v_2+u_1u_2v_1,\,u_2^2v_3-u_2u_3v_2-u_1^2v_3+u_1u_3v_1)=(-u_1u_3v_3+u_3^2v_1+u_1u_2v_2-\\ {}-u_2^2v_1,\,-u_2u_3v_3+u_3^2v_2+u_1^2v_2-u_1u_2v_1,\,-u_2^2v_3+u_2u_3v_2+u_1^2v_3-u_1u_3v_1).$