Vector Product and Angular Velocity

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Vector Product and Angular Velocity

TL;DR

The cross product of two vectors gives a third vector perpendicular to both, with magnitude equal to the area of the parallelogram they span. Angular velocity ω is a vector pointing along the rotation axis, and the linear velocity of any point in a rotating body is v = ω × r, where r is the position vector from the axis.

1. The Mental Model

Imagine a spinning wheel. Every point on it moves in a circle, but that motion isn't random—it's tied to a single rotation axis and a single angular velocity vector. The cross product is the tool that converts "spin rate and direction" into "actual velocity at this spot." It's pure geometry: point your fingers along ω, curl them toward r, and your thumb shows where v points—always perpendicular to both.

2. The Core Material

2.1 The Cross Product: Definition and Geometry

The cross product a × b is a vector perpendicular to both a and b. Its magnitude is:

|a × b| = |a||b| sin θ

where θ is the angle between them. Geometrically, this is the area of the parallelogram with sides a and b.

In component form, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃):

a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

Key properties:
- a × b = −b × a (anti-commutative)
- a × a = 0 (parallel vectors give zero)
- a × (b + c) = a × b + a × c (distributive)
- If a × b = 0 and neither is zero, the vectors are parallel

The right-hand rule tells you the direction: fingers along a, curl toward b, thumb points in the direction of a × b.

2.2 Angular Velocity as a Vector

When a rigid body rotates about a fixed axis, we describe that rotation with angular velocity ω. This is a vector, not just a number.

• Direction: along the rotation axis, determined by the right-hand rule (fingers curl with rotation, thumb points along ω)
• Magnitude: |ω| = dθ/dt (radians per second)

The crucial insight is that every point in the body rotates with the same ω, but their linear velocities differ because they're at different distances from the axis.

2.3 Linking Rotation to Linear Velocity

For a rigid body rotating about a fixed point (or axis through the origin), the linear velocity of a point at position r is:

v = ω × r

This comes from geometry: a point at distance r_⊥ from the axis moves in a circle at speed v = ω r_⊥. The cross product automatically accounts for both the distance and the direction perpendicular to both ω and r.

Example vectors:
- If ω = (0, 0, ω) (rotation about the z-axis) and r = (x, y, z), then:
v = (0, 0, ω) × (x, y, z) = (−ωy, ωx, 0)

Notice: a point on the x-axis moves in the +y direction, a point on the y-axis moves in the −x direction. This is exactly circular motion.

graph TB
    A["Rigid Body Rotating<br/>about Fixed Axis"] --> B["Angular Velocity<br/><b>ω</b><br/>magnitude = dθ/dt<br/>direction: right-hand rule"]
    A --> C["Point at Position<br/><b>r</b> from axis"]
    B --> D["Linear Velocity<br/><b>v</b> = <b>ω</b> × <b>r</b>"]
    C --> D
    D --> E["Speed depends on<br/>distance from axis:<br/>v = ω r sin(90°)"]
    D --> F["Direction is tangent<br/>to circular path"]

2.4 Why the Cross Product Fits

The cross product is not arbitrary—it's the only operation that gives you:
1. A vector perpendicular to the rotation axis
2. Magnitude proportional to distance from the axis
3. Direction that's always tangent to the circular path
4. The correct reversal of sign under reversal of rotation direction

If you tried addition (v = ω + r) or dot product, you'd get nonsense. The cross product is geometry speaking.

3. Worked Example

A disk of radius R rotates about its central axis (the z-axis) with angular velocity ω = (0, 0, 3 rad/s). Find the linear velocity of a point at position r = (2, 0, 0) m.

Step 1: Set up the cross product.
v = ω × r = (0, 0, 3) × (2, 0, 0)

Step 2: Apply the component formula.
v = (0·0 − 3·0, 3·2 − 0·0, 0·0 − 0·2)
v = (0, 6, 0) m/s

Step 3: Interpret.
The point is on the x-axis at distance 2 m from the axis. It moves in the +y direction (perpendicular to both the axis and its position vector) with speed 6 m/s. Check: v = ωr = 3 × 2 = 6 ✓

Now try another point at r = (0, 1, 0) m (on the y-axis, 1 m from the axis):
v = (0, 0, 3) × (0, 1, 0) = (−3, 0, 0) m/s

Speed is 3 m/s (it's half the distance from the axis), and it moves in the −x direction (perpendicular to both axis and position). This is exactly what circular motion predicts.

4. Production Pitfalls & Best Practices

4.1 Real-World Best Practices

1. Always check handedness. Use the right-hand rule or compute components carefully. Reversed vectors lead to reversed forces and torques in dynamics—this breaks everything downstream.

2. Distinguish linear and angular quantities. ω lives in the same space as r, but they measure completely different things. Don't mix units: rad/s for angular, m/s for linear.

3. Use component form for calculations. Mental geometry is good for intuition, but the 3×3 determinant formula (or the explicit component formula) is fast and avoids sign errors.

4. Remember: v = ω × r only when the axis is fixed. In general rigid-body motion, you need the full formula v = v_center + ω × r', where r' is measured from the center of mass.

5. Check dimensional analysis. [ω] = T⁻¹, [r] = L, so [ω × r] = T⁻¹ L = LT⁻¹, which is velocity. ✓

4.2 Common Bugs & Anti-Patterns

Bug 1: Forgetting anti-commutativity.

# WRONG
v = r.cross(omega)  # This gives the opposite direction!

# CORRECT
v = omega.cross(r)  # Right-hand rule: ω first, then r

Bug 2: Using dot product instead of cross product.

# WRONG
v = omega.dot(r)  # This gives a scalar, not a velocity vector!

# CORRECT
v = omega.cross(r)  # This gives a perpendicular vector

Bug 3: Confusing position vector with distance.

# WRONG
r_distance = 3  # This is just a number
v_magnitude = omega_magnitude * r_distance  # Will work, but misses direction

# CORRECT
r_vector = np.array([3, 0, 0])  # Actual position
v_vector = omega.cross(r_vector)  # Full vector with direction

Bug 4: Forgetting to center the rotation.

# WRONG
v = omega.cross(r_world)  # r_world is measured from origin, not axis

# CORRECT
r_from_axis = r_world - axis_position
v = omega.cross(r_from_axis)  # Measure from the actual rotation center

4.3 When to Use What (Decision Matrix)

5. Now Try It

Exercise: Angular Velocity to Linear Velocity

A wheel of radius 0.5 m rotates about a horizontal axis (call it the y-axis) with angular velocity ω = (0, 2, 0) rad/s. Three points on the wheel are located at:
- Point A: r_A = (0.5, 0, 0) m (on the rim, front of wheel)
- Point B: r_B = (0, 0, 0.5) m (on the rim, top of wheel)
- Point C: r_C = (0.25, 0, 0) m (halfway to rim)

For each point, calculate v = ω × r and find:
1. The magnitude of linear velocity (speed)
2. The direction it points

Then check your answers: all speeds should satisfy v = ωr_⊥ where r_⊥ is the perpendicular distance to the y-axis, and all velocities should be perpendicular to both ω and r.

Success looks like: three velocity vectors with correct magnitudes (1 m/s, 1 m/s, 0.5 m/s), directions perpendicular to the axis, and a written explanation of why the speeds follow the pattern ωr_⊥.

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