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 Unit Vector in Direction of Vector:

$\mathbf {\hat r} = \dfrac {\mathbf r} r$

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

$\blacksquare$

Sources

• 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Components of a Vector: $7$. The unit vectors $\mathbf i$, $\mathbf j$, $\mathbf k$