Cross Product#
Definition: Cross Product
The cross product \(\boldsymbol{a} \times \boldsymbol{b}\) of two real column vectors \(\boldsymbol{a} = \begin{bmatrix} a_x & a_y & a_z\end{bmatrix}^\mathsf{T}\) and \(\boldsymbol{b} = \begin{bmatrix} b_x & b_y & b_z\end{bmatrix}^\mathsf{T}\) in \(\mathbb{R}^3\) is defined as the following real column vector:
Theorem: Magnitude of the Cross Product
The Euclidean norm of the cross product \(\boldsymbol{v} \times \boldsymbol{w}\) is given the product of the real sine function and the Euclidean norms of \(v\) and \(w\) as
where \(\theta\) is the angle between \(\boldsymbol{v}\) and \(\boldsymbol{w}\).
Proof
TODO
Theorem: Direction of the Cross Product
If \(v, w \in \mathbb{R}^3\) are real vectors, then the cross product \(v \times w\) is orthogonal to both \(v\) and \(w\) with respect to the dot product:
Tip: Right-Hand Rule
The direction of \(v \times w\) can be determined by the right-hand rule - if you point your index finger in the direction of \(v\) and your middle finger in the direction of \(w\) and then your thumb will point in the direction of \(v \times w\) if you stick it out.
Proof
TODO
Theorem: Self-Product of the Cross Product
The cross product of a vector \(\boldsymbol{v} \in \mathbb{R}^3\) with itself is \(\boldsymbol{0}\).
Proof
TODO
Theorem: Linear Independence via Cross Product
The cross product \(\boldsymbol{v} \times \boldsymbol{w}\) is non-zero if and only if \(\boldsymbol{v}\) and \(\boldsymbol{w}\) are linearly independent.
Proof
TODO
Theorem: Anticommutativity of the Cross Product
The cross product is anticommutative:
Proof
TODO
Theorem: Jacobi Property of the Cross Product (Non-Associativity)
The cross product is not associative but satisfies the Jacobi property:
Proof
TODO
Theorem: Distributivity of the Cross Product
The cross product is distributive over addition:
Proof
TODO
Theorem: Compatibility of the Cross Product
The cross product is compatible with matrix products:
Proof
TODO