Vector Cross Product is Anticommutative/Proof 3
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Theorem
The vector cross product is anticommutative:
- $\forall \mathbf a, \mathbf b \in \R^3: \mathbf a \times \mathbf b = -\left({\mathbf b \times \mathbf a}\right)$
Proof
| \(\ds \mathbf a \times \mathbf b\) | \(=\) | \(\ds \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ a_i & a_j & a_k \\ b_i & b_j & b_k \end {vmatrix}\) | Definition of Vector Cross Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ b_i & b_j & b_k \\ a_i & a_j & a_k \end {vmatrix}\) | Determinant with Rows Transposed | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {\mathbf b \times \mathbf a}\) | Definition of Vector Cross Product |
$\blacksquare$