Vector as Sum of Orthogonal Base Vectors

Theorem
Let $\mathbf v$ be a vector quantity in ordinary $3$-space.

Let $\mathbf i, \mathbf j, \mathbf k$ be orthonormal base vectors.

Then:
 * $\mathbf v = \paren {\mathbf v \cdot \mathbf i} \mathbf i + \paren {\mathbf v \cdot \mathbf j} \mathbf j + \paren {\mathbf v \cdot \mathbf k} \mathbf k$