Cross Product of Vectors

The cross product (also called the vector product) is a way to multiply two vectors in three-dimensional space. Unlike the dot product, the cross product gives another vector, not a number.

This new vector is:

• perpendicular to both original vectors,
• direction decided by the right-hand rule,
• magnitude equal to the area of the parallelogram formed by the vectors.

The magnitude of the cross product shows how far the two vectors spread out from each other. The formula is:

\[ |\vec{u} \times \vec{v}| = |\vec{u}| \, |\vec{v}| \, \sin \theta \]

where \(\theta\) is the angle between the vectors.

If the vectors point in the same or opposite directions (\(\theta = 0^\circ\) or \(180^\circ\)), the cross product is the zero vector.

The direction of the cross product is perpendicular to both vectors and is found using the right-hand rule:

• Point your fingers in the direction of \(\vec{u}\).
• Rotate them toward \(\vec{v}\).
• Your thumb points in the direction of \(\vec{u} \times \vec{v}\).

This rule makes the cross product directional and not symmetric.

The cross product in component form uses a determinant. For:

\[ \vec{u} = \langle a_1, b_1, c_1 \rangle, \quad \vec{v} = \langle a_2, b_2, c_2 \rangle \]

the cross product is:

\[ \vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} \]

Expanding the determinant gives:

\[ \vec{u} \times \vec{v} = \langle b_1c_2 - c_1b_2,\; c_1a_2 - a_1c_2,\; a_1b_2 - b_1a_2 \rangle \]

• Not commutative: \( \vec{u} \times \vec{v} \neq \vec{v} \times \vec{u} \)
• In fact: \( \vec{u} \times \vec{v} = - (\vec{v} \times \vec{u}) \)
• Distributive over addition: \( \vec{u} \times (\vec{v} + \vec{w}) = \vec{u} \times \vec{v} + \vec{u} \times \vec{w} \)
• Zero vector result: If vectors are parallel or one is the zero vector, the cross product is the zero vector.
• Perpendicular vector: The result is perpendicular to both \(\vec{u}\) and \(\vec{v}\).

A very useful application: the cross product of two vectors is the zero vector if and only if the vectors are parallel.

\[ \vec{u} \times \vec{v} = \vec{0} \quad \Longleftrightarrow \quad \vec{u} \text{ and } \vec{v} \text{ are parallel} \]