Vector Quantity as Scalar Product of Unit Vector Quantity

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Theorem

Let $\mathbf a$ be a vector quantity.

Then:

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

where:

$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$
$\mathbf {\hat a}$ denotes the unit vector in the direction $\mathbf a$.


Proof

\(\ds \size {\mathbf {\hat a} }\) \(=\) \(\ds 1\) Definition of Unit Vector
\(\ds \leadsto \ \ \) \(\ds \mathbf {\hat a} \times \size {\mathbf {\hat a} }\) \(=\) \(\ds \mathbf {\hat a}\)
\(\ds \leadsto \ \ \) \(\ds \size {\mathbf a} \times \size {\mathbf {\hat a} } \times \mathbf {\hat a}\) \(=\) \(\ds \size {\mathbf a} \times \mathbf {\hat a}\)
\(\ds \leadsto \ \ \) \(\ds \mathbf a\) \(=\) \(\ds \size {\mathbf a} \mathbf {\hat a}\)

$\blacksquare$


Sources

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