Definition:Vector Cross Product

Definition

Let $\mathbf a$ and $\mathbf b$ be $3$-dimensional vectors, such that:

• $\mathbf a = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k$
• $\mathbf b = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k$

Then the vector cross product, denoted $\mathbf a \times \mathbf b$, is defined as:

$\mathbf a \times \mathbf b = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k\\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end{vmatrix}$

where $\begin{vmatrix} \ldots \end{vmatrix}$ can be understood as a determinant.

More directly:

$\mathbf a \times \mathbf b = (a_j b_k - a_k b_j)\mathbf i - (a_i b_k - a_k b_i)\mathbf j + (a_i b_j - a_j b_i)\mathbf k$

If the vectors are represented as column matrices:

$\mathbf a = \begin{bmatrix} a_i \\ a_j \\ a_k \end{bmatrix} , \mathbf b = \begin{bmatrix} b_i \\ b_j \\ b_k \end{bmatrix}$

we can express the dot product as:

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\(\displaystyle \begin{bmatrix} a_i \\ a_j \\ a_k \end{bmatrix} \times \begin{bmatrix} b_i \\ b_j \\ b_k \end{bmatrix}\)

\(=\)

\(\displaystyle \begin{bmatrix} a_j b_k - a_k b_j \\ a_k b_i - a_i b_k \\ a_i b_j - a_j b_i \end{bmatrix}\)

\(\displaystyle \)

\(\displaystyle \)

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Also see

• Dot Product
• Lagrange's Formula
• Cross Product is Anticommutative
• Cross Product Not Associative

Sources

• For a video presentation of the contents of this page, visit the Khan Academy.