Component of Vector is Scalar Projection on Standard Ordered Basis Element

Theorem
Let $\tuple {\mathbf e_1, \mathbf e_2, \mathbf e_3}$ be the standard ordered basis of Cartesian $3$-space $S$.

Let $\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + a_3 \mathbf e_3$ be a vector quantity in $S$.

Then:
 * $\mathbf a \cdot \mathbf e_i = a_i$

Proof
Using the Einstein summation convention