Vector Cross Product is Anticommutative

Theorem

$\forall \mathbf a, \mathbf b \in \R^3: \mathbf a \times \mathbf b = -\left({\mathbf b \times \mathbf a}\right)$

Complex Plane

The same result holds for complex numbers:

$\forall z_1, z_2 \in \C: z_1 \times z_2 = -\paren {z_2 \times z_1}$

Proof 1

 $\displaystyle \mathbf b \times \mathbf a$ $=$ $\displaystyle \begin{bmatrix} b_i \\ b_j \\ b_k \end{bmatrix} \times \begin{bmatrix} a_i \\ a_j \\ a_k \end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} b_j a_k - a_j b_k \\ b_k a_i - b_i a_k \\ b_i a_j - a_i b_j \end{bmatrix}$ $\displaystyle \mathbf a \times \mathbf b$ $=$ $\displaystyle \begin{bmatrix} a_i \\ a_j \\ a_k \end{bmatrix} \times \begin{bmatrix} b_i \\ b_j \\ b_k \end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} a_j b_k - a_k b_j \\ a_k b_i - a_i b_k \\ a_i b_j - a_j b_i \end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} -\left({a_k b_j - a_j b_k}\right) \\ -\left({a_i b_k - a_k b_i}\right) \\ -\left({a_j b_i - a_i b_j}\right)\end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle -1 \begin{bmatrix} b_j a_k - a_j b_k \\ b_k a_i - b_i a_k \\ b_i a_j - a_i b_j \end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle -\left({\mathbf b \times \mathbf a}\right)$

$\blacksquare$

Proof 2

 $\displaystyle \left({\mathbf a + \mathbf b}\right) \times \left({\mathbf a + \mathbf b}\right)$ $=$ $\displaystyle \mathbf a \times \mathbf a + \mathbf a \times \mathbf b + \mathbf b \times \mathbf a + \mathbf b \times \mathbf b$ Vector Cross Product Operator is Bilinear $\displaystyle 0$ $=$ $\displaystyle 0 + \mathbf a \times \mathbf b + \mathbf b \times \mathbf a + 0$ Cross Product of Vector with Itself is Zero $\displaystyle \mathbf a \times \mathbf b$ $=$ $\displaystyle -\left({\mathbf b \times \mathbf a}\right)$ simplifying

$\blacksquare$

Proof 3

 $\displaystyle \mathbf a \times \mathbf b$ $=$ $\displaystyle \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$ Definition of Vector Cross Product $\displaystyle$ $=$ $\displaystyle -\begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ b_i & b_j & b_k \\ a_i & a_j & a_k \\ \end{vmatrix}$ Determinant with Rows Transposed $\displaystyle$ $=$ $\displaystyle -\left({\mathbf b \times \mathbf a}\right)$ Definition of Vector Cross Product

$\blacksquare$