Like Unit Vectors are Equal
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Theorem
Let $\mathbf a$ and $\mathbf b$ be like vector quantities.
Then:
- $\mathbf {\hat a} = \mathbf {\hat b}$
where $\mathbf {\hat a}$ and $\mathbf {\hat b}$ denote the unit vectors in the direction of $\mathbf a$ and $\mathbf b$.
Proof
By definition of like vector quantities, $\mathbf a$ and $\mathbf b$ have the same direction.
By definition of unit vector, $\mathbf {\hat a}$ and $\mathbf {\hat b}$ are both of magnitude $1$.
Hence the result, by Equality of Vector Quantities.
$\blacksquare$