Component of Vector is Scalar Projection on Standard Ordered Basis Element

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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

\(\ds \mathbf a \cdot \mathbf e_i\) \(=\) \(\ds a_j \cdot \mathbf e_j \cdot \mathbf e_i\)
\(\ds \) \(=\) \(\ds a_j \delta_{i j}\) Dot Product of Orthonormal Basis Vectors
\(\ds \) \(=\) \(\ds a_i\)

$\blacksquare$


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