Hard

Vector (Cross) Product of Vectors

Simple Explanation

The vector (cross) product multiplies two vectors together to produce a NEW vector, perpendicular to both original vectors. Its magnitude depends on how perpendicular the two original vectors are to each other, and its direction follows the right-hand rule.

Why Do We Need It?

The cross product is how physics defines torque and angular momentum — both are naturally perpendicular to the plane formed by the two vectors that produce them.

Formula

Vector (Cross) Product

|A × B| = AB sinθ

The vector product of two vectors gives a new vector, perpendicular to both original vectors, with a magnitude that depends on how perpendicular the two original vectors are to each other.

A × B: — The vector (cross) product, itself a vector
A, B: — Magnitudes of the two vectors
θ: — The angle between the two vectors

When to use it: Use when combining two vectors should produce a new vector perpendicular to both, such as calculating torque (r × F) or angular momentum.

Worked Example

Calculating a vector product magnitude

A force of 12 N is applied 90° to a 0.4 m lever arm. Find the magnitude of the vector product (torque).

Why Does This Work?

When θ = 90° (vectors fully perpendicular), sinθ = 1, so the cross product reaches its maximum possible magnitude. When θ = 0° (vectors parallel), sinθ = 0, so the cross product is zero — a force applied directly along the lever arm produces no turning effect, matching physical intuition.

Real-Life Example

Why pushing a door near its hinge is hard

Pushing a door open right next to its hinges takes much more force than pushing near the handle, even to produce the same turning effect.

Torque is a cross product (r × F) — with a very small r (distance from hinge), a much larger force F is needed to produce the same magnitude of torque, which is exactly why door handles are placed far from the hinge.

Practice

Two vectors of magnitude 5 and 4 act at 30° to each other (sin30° = 0.5). Find the magnitude of their vector product.

Hard

Common mistake

Confusing the cross product with the dot product — the cross product uses SINE (maximum at 90°) and produces a VECTOR, while the dot product uses COSINE (maximum at 0°) and produces a scalar.

Quick Review

|A × B| = AB sinθ produces a vector, perpendicular to both original vectors.
Maximum when vectors are perpendicular (θ=90°); zero when parallel (θ=0°).
Used to define torque (r × F) and angular momentum.