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 $S$.

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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

\(\ds \)

\(=\)

\(\ds a_i\)

$\blacksquare$

Sources

• 1992: Frederick W. Byron, Jr. and Robert W. Fuller: Mathematics of Classical and Quantum Physics ... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product