Vector Cross Product is Anticommutative

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Theorem

The vector cross product is anticommutative:

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


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

$\blacksquare$


Sources

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