Cross Product in LaTeX

Write the cross product with \times between two vectors in math mode: \mathbf{a} \times \mathbf{b} or \vec{a} \times \vec{b}. To show how it is computed, put the unit vectors and components in a vmatrix determinant.

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Cross product notation

\mathbf{a} \times \mathbf{b}
$a \times b$
Bold vectors with the times sign.
\vec{a} \times \vec{b}
$\vec{a} \times \vec{b}$
Arrow vectors with the times sign.
\mathbf{i}, \mathbf{j}, \mathbf{k}
$i, j, k$
Bold unit vectors.
\hat{\imath}, \hat{\jmath}, \hat{k}
$\hat{ı}, \hat{ȷ}, \hat{k}$
Hat unit vectors; \imath and \jmath are dotless i and j.
\begin{vmatrix} a & b \\ c & d \end{vmatrix}
$| \begin{matrix} a & b \\ c & d \end{matrix} |$
Determinant bars for the component formula (amsmath).
\|\mathbf{a} \times \mathbf{b}\|
$∥ a \times b ∥$
Magnitude of the cross product.
\mathbf{a} \wedge \mathbf{b}
$a \land b$
Wedge notation, used in some texts and for the exterior product.

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Change the formula and the preview redraws as you type. Copy the LaTeX, download a PNG or SVG, copy it for Word, or use Edit here on any example below to load it.

LaTeX

Templates

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Cross product examples

Each result below is drawn from the LaTeX under it. Copy the code, edit it on this page, or open it in the full editor.

Determinant form

a \times b = | \begin{matrix} i & j & k \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} |

LaTeX

\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

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Component form

a \times b = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})

LaTeX

\mathbf{a} \times \mathbf{b} = (a_2 b_3 - a_3 b_2,\; a_3 b_1 - a_1 b_3,\; a_1 b_2 - a_2 b_1)

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Magnitude

∥ a \times b ∥ = ∥ a ∥ ∥ b ∥ \sin θ

LaTeX

\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \, \|\mathbf{b}\| \sin\theta

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Anticommutativity

a \times b = - b \times a

LaTeX

\mathbf{a} \times \mathbf{b} = -\,\mathbf{b} \times \mathbf{a}

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Unit vectors
$\hat{ı} \times \hat{ȷ} = \hat{k}$
LaTeX
```
\hat{\imath} \times \hat{\jmath} = \hat{k}
```
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Torque
$\vec{τ} = \vec{r} \times \vec{F}$
LaTeX
```
\vec{\tau} = \vec{r} \times \vec{F}
```
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Scalar triple product
$a \cdot (b \times c)$
LaTeX
```
\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})
```
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The determinant form

The usual way to show the computation is a 3×3 determinant: unit vectors in the first row, the components of a and b below. vmatrix (from amsmath) draws the vertical bars at the full height of the matrix; hand-typed | characters would stay one line tall.

LaTeX

\[
\mathbf{a} \times \mathbf{b} =
\begin{vmatrix}
  \mathbf{i} & \mathbf{j} & \mathbf{k} \\
  a_1 & a_2 & a_3 \\
  b_1 & b_2 & b_3
\end{vmatrix}
\]

Hats on i and j

A hat over a normal i or j sits above the dot. Use the dotless letters \imath and \jmath under accents: \hat{\imath}, \hat{\jmath}. k has no dot, so \hat{k} is fine.

Other notations

× is the standard symbol. Some older and European texts write a ∧ b (\wedge) for the vector product; in modern mathematics ∧ usually means the exterior (wedge) product. Define which one you mean if your readers may differ.

Common mistakes

The letter x is a variable, not an operator, and gets no spacing.
Instead of
```
a x b
```
Write
```
\mathbf{a} \times \mathbf{b}
```
The hat collides with the dot of i and j; the dotless forms look right.
Instead of
```
\hat{i} \times \hat{j}
```
Write
```
\hat{\imath} \times \hat{\jmath}
```
Typed bars stay one line tall; vmatrix stretches them to the matrix.
Instead of
```
| \begin{matrix} … \end{matrix} |
```
Write
```
\begin{vmatrix} … \end{vmatrix}
```
\vec is a short arrow sized for one letter; \overrightarrow spans both.
Instead of
```
\vec{AB} \times \vec{AC}
```
Write
```
\overrightarrow{AB} \times \overrightarrow{AC}
```

In Overleaf, MathJax and Word

Overleaf and pdfLaTeX
\times, \vec, \mathbf, \imath and \jmath are built in; vmatrix needs amsmath.
MathJax and KaTeX
MathJax and KaTeX render all of it, vmatrix included, without loading packages.
Microsoft Word
Write the formula here and use Copy for Word; the determinant pastes as an editable Word equation with bars around the matrix.