Components of Vector in Terms of Dot Product

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

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 $\mathbf a$ be expressed in component form:
 * $\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$

Then: