Scalar Multiplication by Zero gives Zero Vector

Theorem

Let $\mathbf a$ be a vector quantity.

Let $0 \mathbf a$ denote the scalar product of $\mathbf a$ with $0$.

Then:

$0 \mathbf a = \bszero$

where $\bszero$ denotes the zero vector.

Proof

By definition of scalar product:

$\size {0 \mathbf a} = 0 \size {\mathbf a}$

where $\size {\mathbf a}$ denotes the magnitude of $\mathbf a$.

Thus:

$\size {0 \mathbf a} = 0$

That is: $0 \mathbf a$ is a vector quantity whose magnitude is zero.

Hence, by definition, $0 \mathbf a$ is the zero vector.

$\blacksquare$