Vector as Sum of Orthogonal Base Vectors

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

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

Then:
 * $\mathbf v = \left({\mathbf v \cdot \mathbf i}\right) \mathbf i + \left({\mathbf v \cdot \mathbf j}\right) \mathbf j + \left({\mathbf v \cdot \mathbf k}\right) \mathbf k$