Cross Product | Notion

Table of Contents

</aside>

Definition

Cross product is a vector multiplication operation that results in another vector, not a scalar (unlike the dot product).

This operation denotes ($\times$), unlike the denotation of dot product ($·$)

Here is $\mathbf{u}=(u_x, u_y, u_z)$ & $\mathbf{v} = (v_x, v_y, v_z)$

Then the cross product is computed like so.

$$ \mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_y v_z - u_z v_y \\ u_z v_x - u_x v_z \\ u_x v_y - u_y v_x \end{pmatrix}

Result $\mathbf{w}=\mathbf{u}\times\mathbf{v}$ is orthogonal to both $\mathbf{u}$ and $\mathbf{v}$.

Geometric Interpretation

Magnitude

Blue & Red Vector | Cross Product | Area of Parallelogram

Blue & Red Vector | Cross Product | Area of Parallelogram

The magnitude of the cross product vector $\|\mathbf{u}\times\mathbf{v}\|$ represents the area of the parallelogram spanned by $\mathbf{u}$ and $\mathbf{v}$.

$$ \|\mathbf{u}\times\mathbf{v}\|=\|\mathbf{u}\|\|\mathbf{v}\|\sin\theta $$

$\theta$ is the angle between the two vectors.
If the vectors are parallel ($θ$ = $0°$ or $180°$), the cross product $= 0$,
If the vectors are orthogonal ($θ$ = $90°$), the magnitude is at its maximum.

Example

Here is $\mathbf{u}=(2,0,1)$, $\mathbf{v}=(1,0,3)$.

Cross product

$$ \mathbf{u}\times\mathbf{v} = \begin{pmatrix} (0)(3) - (1)(0) \\ (1)(1) - (2)(3) \\ (2)(0) - (0)(1) \end{pmatrix} = \begin{pmatrix} 0 \\ -5 \\ 0 \end{pmatrix} $$
- The result is orthogonal to both input vectors, pointing in the $-Y$ direction.
Area of the parallelogram

$$ \|\mathbf{u}\| = \sqrt{2^2 + 0^2 + 1^2} = \sqrt{5} $$

$$ \|\mathbf{v}\| = \sqrt{1^2 + 0^2 + 3^2} = \sqrt{10} $$
- Finding $\theta$
  
  $$ \cos\theta = \frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|} = \frac{5}{\sqrt{50}} $$
  
  $$ \theta = \arccos(\frac{5}{\sqrt{50}}) = 45° $$
$$ \|\mathbf{u}\times\mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\sin\theta = 5 $$
- From this, determine that the area of the parallelogram is $5$.
- In other words, the magnitude of the cross product vector equals the area of the parallelogram formed by the two vectors.

Direction

The direction of the resulting vector $\mathbf{w}$ = $\mathbf{u}$ × $\mathbf{v}$ can be easily found using the right-hand rule.

Point your fingers in the direction of $\mathbf{u}$, curl them toward $\mathbf{v}$, and your thumb points in the direction of $\mathbf{w}$.

Why Cross Product follows “Right-Hand-Rule”

Why does the "right-hand-rule" for vector cross-products work?

Summary : The right-hand rule is nothing more than a matter of convention. </aside>

Properties

Anti-commutativity

This property shows that $\mathbf{u} \times \mathbf{v} \neq \mathbf{v} \times \mathbf{u}$ in general. so that we say that the cross product is anti-commutative.

$$ \mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u}) $$

Determine the vector returned by the cross product by the left-hand-thumb rule.

Parallel Vectors

If $\|\sin\theta\|=0$ (i.e., $\theta=0°$ or $\theta=180°$), then $\|\mathbf{u}\times\mathbf{v}\|=0$

It means parallel vectors produce a zero vector.

Orthogonalization with Cross Product

Cross product allows to find a vector orthogonal to two 3D vectors.

Here is $\mathbf{u}$ and $\mathbf{v}$.

Find a vector $\mathbf{w}$ orthogonal to both.

$$ \mathbf{w} = \mathbf{u} \times \mathbf{v} $$

$\mathbf{w} \cdot \mathbf{u} = 0$ (orthogonal to $\mathbf{u}$)
$\mathbf{w} \cdot \mathbf{v} = 0$ (orthogonal to $\mathbf{v}$)

This property is particularly useful.

Creating coordinate systems
Finding normal vectors to planes
Generating basis vectors

Practice : Graph

3D Graph

https://www.desmos.com/3d/wyoywrymn9

</aside>

Elements

$\mathbf{u}$ Red (vector)