Components of Vector in Terms of Dot Product

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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Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $2$. The Scalar Product: $(2.9)$