Question

Suppose u and v are nonzero vectors in 3-space, where u=(u1, u2, u3) and v=(v1, v2, v3). Prove that u x v is perpendicular to both u and v by making use of the dot product.

Answer 1

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

$u\cdot c=u_1(u_2v_3-u_3v_2)-u_2(u_1v_3-u_3v_1)+u_3(u_1v_2-u_2v_1)=0$

Which means u is perpendicular to c.

$v\cdot c=v_1(u_2v_3-u_3v_2)-v_2(u_1v_3-u_3v_1)+v_3(u_1v_2-u_2v_1)=0$

Which means v is perpendicular to c.