Q.Show that $(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=2(\vec{a}\times\vec{b})$.

Concept: Vector Magnitude Properties — specifically, the distributive and anti-commutative properties of the cross product.

Step 1: Expand using distributivity:

Step 2: Recall that for any vector, $\vec{v} \times \vec{v} = \vec{0}$. Also, $\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})$. Substituting: …

This is a classic vector identity that looks intimidating at first but collapses beautifully once you apply two simple rules: cross product distributes over addition, and any vector crossed with itself gives zero.

The cross product is distributive over vector addition, meaning:

and similarly from the other side. This is the only tool we need — no geometry, no components, no determinants.

Let’s work through it step by step.

1. Expand the left-hand side using distribution. Treat $(\vec{a} - \vec{b})$ as one vector and $(\vec{a} + \vec{b})$ as the other. Distribute from left to right:

2. Distribute again inside each term. For the first term:

For the second term:

3. Apply the zero self-cross property. Any vector crossed with itself is the zero vector: $\vec{a} \times \vec{a} = \vec{0}$ and $\vec{b} \times \vec{b} = \vec{0}$. So the expression becomes:

which simplifies to: …

Method: Expanding a Cross Product of Two Sums

Use distributivity to expand cross products just like FOIL — but keep every factor's left-right order, because the cross product is anti-commutative.

Steps

Step 1: Distribute term by term, preserving order.

Step 2: Kill the self-cross terms. …

Common Mistakes

Mistake 1: Treating the cross product as commutative.

Why it's wrong: $\vec{b}\times\vec{a}=-(\vec{a}\times\vec{b})$, so order matters; ignoring the sign makes the two terms cancel to $\vec{0}$ instead of adding. Correct approach: apply anti-commutativity, so they combine to $2(\vec{a}\times\vec{b})$.

Mistake 2: Forgetting $\vec{a}\times\vec{a}=\vec{0}$.

Why it's wrong: leaving self-cross terms in the expansion adds spurious pieces. Correct approach: any vector crossed with itself is the zero vector, so those terms drop. …