Q.If a⃗ is a non-zero vector then |a⃗ × a⃗| is equal to:

Cross Product of Parallel Vectors

Imagine you're trying to open a door. You push on the handle — that force works because it's perpendicular to the door. If you push along the door (parallel to its surface), nothing happens. The cross product measures exactly this "perpendicular effectiveness" between two vectors.

When two vectors are parallel, they point in exactly the same direction (or exactly opposite). There is no "perpendicular component" between them, so the cross product — which captures that perpendicular interaction — must be zero.

The Intuition

Take two parallel vectors $\vec{a}$ and $\vec{b}$, two arrows lying along the same line. No matter how you rotate them, you cannot get one to point "across" the other. The area of the parallelogram they span is zero — a degenerate, flat shape. The cross product gives the vector perpendicular to both, with magnitude equal to that area. Since the area is zero, the cross product is the zero vector.

The Precise Statement

If $\vec{a}$ and $\vec{b}$ are parallel (i.e. $\vec{b} = k\vec{a}$ for some scalar $k$), then:

The converse is also true: if the cross product of two non-zero vectors is zero, they must be parallel (or anti-parallel).

Why This Matters in Exams

This is a quick check for parallelism: compute a cross product and get zero, and you immediately know the vectors are collinear. It's also used in proofs — for example, showing two lines are parallel by taking the cross product of their direction vectors.

Quick Example …