Scalar Multiplication by Zero gives Zero Vector
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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.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Fundamental Definitions: $2.$