Cross Product

The cross product is a Vector operation that only applies to two vectors in ${\mathbb{R}}^3$ , i.e. vectors in 3D Real Space. It returns a vector perpendicular to both vectors.

The cross product is only defined in ${\mathbb{R}}^3$

Definition §

Formula §

$$\vec{u}\times \vec{v}=(u_2{v}_3-u_3{v}_2)\hat{i}+(u_3{v}_1-u_1{v}_3)\hat{j}+(u_1{v}_2-u_2{v}_1)\hat{k}$$

$$\vec{u}\times \vec{v}=\Vert u\Vert \Vert v\Vert sin(\theta )\hat{n}$$

• $\vec{u},\vec{v}$ = Two vectors in ${\mathbb{R}}^3$
• $\hat{i},\hat{j},\hat{k}$ = Unit vectors in ${\mathbb{R}}^3$
• $\theta$ = Angle between $\vec{u}$ and $\vec{v}$
• $\hat{n}$ = Unit vector perpendicular to both $\vec{u}$ and $\vec{v}$

(More common) Matrix Formula §

$$\vec{u}\times \vec{v}=\det (\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right])=\hat{i}\det (\left[\begin{array}{cc}{u}_2& u_3\\ v_2& v_3\end{array}\right])-\hat{j}\det (\left[\begin{array}{cc}{u}_1& u_3\\ v_1& v_3\end{array}\right])+\hat{k}\det (\left[\begin{array}{cc}{u}_1& u_2\\ v_1& v_2\end{array}\right])$$

• $\det ()$ = Matrix Determinant
• $\vec{u},\vec{v}$ = Can be written as row matrices

Properties §

Let $\vec{u},\vec{v},\vec{w}\in {\mathbb{R}}^3$ be two vectors and $k\in \mathbb{R}$ be a scalar.

1. Anti-Commutative: Switching the vectors in a cross product returns a negative of the original cross product.$$\vec{v} \times \vec{u} = -(\vec{u} \times \vec{v})

2. **Distributive**: The cross product can be expanded. $$(\vec{u} + \vec{v}) \times \vec{w} = \vec{u} \times \vec{w} + \vec{v} \times \vec{w}

4. Non-Associative: Just like the Dot Product, we can’t switch around the parenthesis in the cross product.$$\vec{u} \times (\vec{v} + \vec{w}) \neq (\vec{u} + \vec{v}) \times \vec{w} \ \ \ \ \ \ \ \vec{u} \times (\vec{v} + \vec{w}) = \vec{u} \times \vec{v} + \vec{u}\times \vec{w}

5. **Obeys scalar multiplication**: We can 'pull out' the scalar from a cross product operation:$$(k\vec{u})\times \vec{v} = \vec{u} \times (k\vec{v})= k(\vec{u} \times \vec{v})

6. The cross product of a vector against itself is always zero.$$\vec{v} \times \vec{v} = 0

7. The cross product of parallel vectors is always zero.$$\vec{u} \times \vec{v} \iff \vec{u} \parallel\vec{v}

Geometry §

#todo

The cross product of two vectors can be thought of as the signed area of a parallelogram created from those two vectors: