Cross product

For the product of two vectors in general, see Vector multiplication.

The cross product is a binary operation on two vectors in three-dimensional Euclidean space (denoted as \(\mathbb{R}^3\)). It results in a vector that is perpendicular to both of the vectors being multiplied and thus normal to the plane containing them. The cross product is denoted by the symbol \(\times\).

Definition

Given two vectors \(\mathbf{a}\) and \(\mathbf{b}\) in \(\mathbb{R}^3\), their cross product \(\mathbf{a} \times \mathbf{b}\) is defined as: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \] where \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the unit vectors in the direction of the x, y, and z axes, respectively, and \(a_1, a_2, a_3\) and \(b_1, b_2, b_3\) are the components of vectors \(\mathbf{a}\) and \(\mathbf{b}\).

The result is: \[ \mathbf{a} \times \mathbf{b} = (a_2 b_3 - a_3 b_2)\mathbf{i} - (a_1 b_3 - a_3 b_1)\mathbf{j} + (a_1 b_2 - a_2 b_1)\mathbf{k} \]

Properties

• **Anticommutativity**: \(\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})\)
• **Distributivity**: \(\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}\)
• **Scalar multiplication**: \((c\mathbf{a}) \times \mathbf{b} = c(\mathbf{a} \times \mathbf{b})\) where \(c\) is a scalar.
• **Zero vector**: \(\mathbf{a} \times \mathbf{a} = \mathbf{0}\)

Geometric Interpretation

The magnitude of the cross product \(\mathbf{a} \times \mathbf{b}\) is given by: \[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin(\theta) \] where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\). The direction of \(\mathbf{a} \times \mathbf{b}\) is given by the right-hand rule.

Applications

The cross product is used in various fields such as physics, engineering, and computer graphics. Some common applications include:

• Calculating the torque exerted by a force.
• Finding the normal vector to a surface.
• Determining the area of a parallelogram formed by two vectors.

Related Concepts

• Dot product
• Vector space
• Linear algebra
• Multivariable calculus

See Also

• Vector calculus
• Triple product
• Determinant
• Orthogonal vectors