Components of Vector in Terms of Dot Product

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Theorem

Let $\mathbf a$ be a vector quantity embedded in Cartesian $3$-space $S$.

Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.

Let $\mathbf a$ be expressed in component form:

$\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$


Then:

\(\ds x\) \(=\) \(\ds \mathbf a \cdot \mathbf i\)
\(\ds y\) \(=\) \(\ds \mathbf a \cdot \mathbf j\)
\(\ds z\) \(=\) \(\ds \mathbf a \cdot \mathbf k\)


Proof


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