Dot Product with Zero Vector is Zero

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Theorem

Let $\mathbf u$ be a vector quantity.

Let $\cdot$ denote the dot product operator.

Then:

$\mathbf u \cdot \mathbf 0 = 0$

where $\mathbf 0$ denotes the zero vector.


Proof

By definition of dot product:

\(\ds \mathbf u \cdot \mathbf 0\) \(=\) \(\ds \norm {\mathbf u} \norm {\mathbf 0} \cos \theta\) Definition of Dot Product
\(\ds \) \(=\) \(\ds \norm {\mathbf u} \times 0 \times \cos \theta\)
\(\ds \) \(=\) \(\ds 0\)

$\blacksquare$


Sources

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