Components of 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 $x$, $y$ and $z$ be the components of $\mathbf r$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions respectively.

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

Then:

Proof

 * Vector-Components-in-3-Space.png

By definition, the direction cosines are the cosines of the angles that $\mathbf r$ makes with the coordinate axes.

By definition of the components of $\mathbf r$:
 * $\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$

Thus:


 * $\mathbf r = r \cos \alpha \mathbf i + r \cos \beta \mathbf j + r \cos \gamma \mathbf k$

and the result follows.