Unit Vector in terms of Direction Cosines

Theorem
Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space.

Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.

Let $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ be the direction cosines of $\mathbf r$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively.

Let $\mathbf {\hat r}$ denote the unit vector in the direction of $\mathbf r$.

Then:
 * $\mathbf {\hat r} = \paren {\cos \alpha} \mathbf i + \paren {\cos \beta} \mathbf j + \paren {\cos \gamma} \mathbf k$

Proof
From Components of Vector in terms of Direction Cosines:


 * $(1): \quad \mathbf r = r \cos \alpha \mathbf i + r \cos \beta \mathbf j + r \cos \gamma \mathbf k$

where $r$ denotes the magnitude of $\mathbf r$, that is:
 * $r := \size {\mathbf r}$

By definition of unit vector:
 * $\mathbf {\hat r} = \dfrac {\mathbf r} r$

The result follows by multiplication of both sides of $(1)$ by $\dfrac 1 r$.