Lesson 5: Cross Product & Triple Product

1. Cross Product (3D Only)

The cross product \(\vec{a} × \vec{b}\) is a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\).

Magnitude = \(\|\vec{a}\| \|\vec{b}\| \sin \theta\) (area of parallelogram)

Direction: right-hand rule (fingers from a to b, thumb shows direction of result).

Formula:

\[ \vec{a} × \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = \begin{pmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{pmatrix} \]

2. Properties of Cross Product

• Anti-commutative: \(\vec{a} × \vec{b} = - \vec{b} × \vec{a}\)
• \(\vec{a} × \vec{a} = \vec{0}\)
• Perpendicular to both input vectors
• Used in torque, angular momentum, normal vectors in 3D graphics

3. Scalar Triple Product

The scalar triple product \(\vec{a} \cdot (\vec{b} × \vec{c})\) gives the volume of the parallelepiped formed by the three vectors.

It is zero if the vectors are coplanar (lie in the same plane).

Formula: equals the determinant of the matrix formed by the three vectors as rows or columns.