Vector (Cross) Product

Definition of the Vector Product

The vector product (or cross product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is:

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

where:

• $\theta \in [0, \pi]$ is the angle from $\mathbf{a}$ to $\mathbf{b}$
• $\hat{\mathbf{n}}$ is a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$
• The direction of $\hat{\mathbf{n}}$ follows the right-hand rule: curl fingers from $\mathbf{a}$ to $\mathbf{b}$, thumb points along $\hat{\mathbf{n}}$

The result is a vector (hence the name).

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Properties

Algebraic properties:

• Anti-commutative: $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$
• Distributive: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$
• Self-product: $\mathbf{a} \times \mathbf{a} = \mathbf{0}$
• Not associative: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$

Geometric interpretation: $|\mathbf{a} \times \mathbf{b}| = \text{area of parallelogram formed by } \mathbf{a} \text{ and } \mathbf{b}$

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Component Form (Determinant)

The cross product can be computed using a determinant:

$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$

Expanding:

$\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y)\hat{\mathbf{i}} + (a_z b_x - a_x b_z)\hat{\mathbf{j}} + (a_x b_y - a_y b_x)\hat{\mathbf{k}}$

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Standard Basis Products

Using the right-hand rule:

$\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}}, \quad \hat{\mathbf{j}} \times \hat{\mathbf{k}} = \hat{\mathbf{i}}, \quad \hat{\mathbf{k}} \times \hat{\mathbf{i}} = \hat{\mathbf{j}}$

And by anti-commutativity:

$\hat{\mathbf{j}} \times \hat{\mathbf{i}} = -\hat{\mathbf{k}}, \quad \hat{\mathbf{k}} \times \hat{\mathbf{j}} = -\hat{\mathbf{i}}, \quad \hat{\mathbf{i}} \times \hat{\mathbf{k}} = -\hat{\mathbf{j}}$

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Applications

Normal Vector to a Plane

Given two vectors in a plane, their cross product gives a normal vector:

$\mathbf{n} = \mathbf{a} \times \mathbf{b}$

Equation of a Line Through Two Points

For points $\mathbf{a}$ and $\mathbf{b}$, the line equation is:

$(\mathbf{r} - \mathbf{a}) \times (\mathbf{b} - \mathbf{a}) = \mathbf{0}$

This encodes: any point $\mathbf{r}$ on the line has $\mathbf{r} - \mathbf{a}$ parallel to $\mathbf{b} - \mathbf{a}$.

Shortest Distance: Point to Line

Distance from point $\mathbf{q}$ to line through points $\mathbf{a}$, $\mathbf{b}$:

$d = \frac{|(\mathbf{q} - \mathbf{a}) \times (\mathbf{b} - \mathbf{a})|}{|\mathbf{b} - \mathbf{a}|}$

This is the magnitude of the perpendicular component.

Rotational Motion

The velocity of a point at position $\mathbf{r}$ rotating with angular velocity $\boldsymbol{\omega}$:

$\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$

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Triple Products

Scalar Triple Product

$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix} = [\mathbf{a}, \mathbf{b}, \mathbf{c}]$

Properties:

• Cyclic permutation (even): value unchanged
  • $[\mathbf{a}, \mathbf{b}, \mathbf{c}] = [\mathbf{b}, \mathbf{c}, \mathbf{a}] = [\mathbf{c}, \mathbf{a}, \mathbf{b}]$
• Odd permutation: sign changes
  • $[\mathbf{a}, \mathbf{c}, \mathbf{b}] = -[\mathbf{a}, \mathbf{b}, \mathbf{c}]$
• Geometric meaning: volume of parallelepiped formed by $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$
• Coplanar vectors: if $[\mathbf{a}, \mathbf{b}, \mathbf{c}] = 0$, the vectors lie in the same plane

Vector Triple Product

$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$

This is the BAC-CAB rule (mnemonic: “BAC minus CAB”).

Warning: Cross product is not associative: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$

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Why the Cross Product Only Exists in 3D

The cross product as defined here is specific to three dimensions. In 2D, a perpendicular “vector” would have to point out of the plane (making it essentially a scalar). In 4D and higher, the concept generalises differently (wedge product / exterior algebra).

This is why the cross product is sometimes called the “vector product” — it produces a vector only because we happen to work in 3D.