Components of Vector in Terms of Dot Product
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This theorem requires a proof.
Not sure how to start this without it being circular
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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
Not sure how to start this without it being circular
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a 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: $(2.9)$