Cross Product of Vector with Itself is Zero

Theorem

Let $\mathbf x$ be a vector in a vector space of $3$ dimensions:

$\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$

Then:

$\mathbf x \times \mathbf x = \mathbf 0$

where $\times$ denotes vector cross product.

Proof 1

\(\ds \mathbf x \times \mathbf x\)

\(=\)

\(\ds \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ x_i & x_j & x_k \\ x_i & x_j & x_k \\ \end {vmatrix}\)

Definition of Vector Cross Product

\(\ds \)

\(=\)

\(\ds \mathbf 0\)

Square Matrix with Duplicate Rows has Zero Determinant

$\blacksquare$

Proof 2

By definition, a vector is parallel to itself.

The result follows from Cross Product of Parallel Vectors.

$\blacksquare$