## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$0$
Suppose that the two vectors can be represented as: $v=v_1i+v_2j+v_3k$ and $w=w_1i+w_2j+w_3k$, then their cross product of such vectors can be obtained in the form of determinate as : $v \times w=\begin{vmatrix} i & j & k \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{vmatrix}=(v_2w_3-v_3w_2)i-(v_1w_3-v_3w_1)j+(v_1w_2-v_2w_1)k$ Here,we have the cross product of two similar vectors as : $v \times v =0$ Because the cross product of a vector by itself is always zero. The cross product of any vector with $0$ is the zero vector, so $u \times (v \times v) =u \times 0=0$