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Section 12.4: Cross Product

Example: Cross product with parallel vectors

EXAMPLE 2 Show that \(\mathbf{a} \times \mathbf{a} = \mathbf{0}\) for any vector a in \(V_3\).

SOLUTION If \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\), then \[ \mathbf{a} \times \mathbf{a} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ a_1 & a_2 & a_3 \end{vmatrix} = (a_2a_3 - a_3a_2)\mathbf{i} - (a_1a_3 - a_3a_1)\mathbf{j} + (a_1a_2 - a_2a_1)\mathbf{k} = 0\mathbf{i} - 0\mathbf{j} + 0\mathbf{k} = \mathbf{0} \]