Cross Product

The cross product (or vector product) is a 3D operation that takes two vectors and produces a third vector perpendicular to both. It’s essential for computing torques, angular momentum, and surface normals in robotics.

What is the Cross Product?

The cross product of two vectors produces a vector:

$\vec{a} \times \vec{b} = \begin{bmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{bmatrix}$

Geometric Interpretation

The magnitude of the cross product equals:

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin(\theta)$

where $\theta$ is the angle between vectors.

Direction: The result is perpendicular to both $\vec{a}$ and $\vec{b}$, following the right-hand rule.

Interactive Exploration

Explore the cross product in 3D. Notice how the red vector is always perpendicular to both blue and green vectors:

Right-Hand Rule

To find the direction of $\vec{a} \times \vec{b}$:

1. Point your fingers along $\vec{a}$
2. Curl them toward $\vec{b}$
3. Your thumb points along $\vec{a} \times \vec{b}$

Geometric Meaning

The magnitude $|\vec{a} \times \vec{b}|$ equals the area of the parallelogram formed by vectors $\vec{a}$ and $\vec{b}$.

Try enabling “Show parallelogram” in the interactive demo to see this!

Special cases:

• Parallel vectors: $\vec{a} \times \vec{b} = \vec{0}$ (zero vector)
• Perpendicular vectors: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|$ (maximum magnitude)

Torque Calculation

Torque ($\vec{\tau}$) is the rotational equivalent of force. It’s calculated using the cross product:

$\vec{\tau} = \vec{r} \times \vec{F}$

where:

• $\vec{r}$ is the position vector from the pivot point to where force is applied
• $\vec{F}$ is the applied force

Properties of Cross Product

1. Anti-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$
2. Distributive: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$
3. Scalar multiplication: $(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})$
4. Self cross product: $\vec{a} \times \vec{a} = \vec{0}$

Perpendicularity Check

If $\vec{c} = \vec{a} \times \vec{b}$, then:

$\vec{a} \cdot \vec{c} = 0 \quad \text{and} \quad \vec{b} \cdot \vec{c} = 0$

The result is perpendicular to both input vectors!

Surface Normals

A normal vector is perpendicular to a surface. Given two vectors in the plane, their cross product gives the normal:

Angular Velocity and Momentum

In robotics, angular velocity ($\vec{\omega}$) and angular momentum ($\vec{L}$) involve cross products:

$\vec{v} = \vec{\omega} \times \vec{r}$

This gives the linear velocity of a point at position $\vec{r}$ on a rotating body.

Coordinate Frame Construction

Given one axis direction, we can construct a full coordinate frame using cross products:

Given: z-axis direction ẑ
Choose: arbitrary vector v (not parallel to ẑ)
Compute:
  x̂ = v × ẑ (normalized)
  ŷ = ẑ × x̂ (normalized)

This creates an orthonormal frame (mutually perpendicular unit vectors).

Robotics Applications

1. Joint Torque Computation

Essential for robot dynamics and control.

2. Collision Detection

Check if objects intersect by testing normal vectors.

3. Camera and Sensor Orientation

Define coordinate frames for vision systems.

4. Path Planning

Compute surface normals for navigation on irregular terrain.

5. Jacobian Matrix

Relates joint velocities to end-effector velocities (covered in Module 5).

Key Takeaways

• Cross product produces a vector perpendicular to both inputs
• Magnitude equals area of parallelogram: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin(\theta)$
• Direction follows right-hand rule
• Essential for torque: $\vec{\tau} = \vec{r} \times \vec{F}$
• Used for surface normals and coordinate frame construction
• Anti-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$

Practice Problems

1. Calculate $\vec{a} \times \vec{b}$ for $\vec{a} = (1, 0, 0)$ and $\vec{b} = (0, 1, 0)$.

2. Verify that $(1, 2, 3) \times (4, 5, 6)$ is perpendicular to both input vectors using dot product.

3. A force of $(5, 0, 0)$ N is applied at position $(0, 0.2, 0)$ m. What’s the torque about the origin?

4. Find the unit normal to a surface spanned by $(1, 0, 0.1)$ and $(0, 1, 0.2)$.

5. If $|\vec{a}| = 3$, $|\vec{b}| = 4$, and the angle between them is 30°, what is $|\vec{a} \times \vec{b}|$?