Scalar Multiplication of Vectors is Distributive over Vector Addition
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Theorem
Let $\mathbf a, \mathbf b$ be a vector quantities.
Let $m$ be a scalar quantity.
Then:
- $m \paren {\mathbf a + \mathbf b} = m \mathbf a + m \mathbf b$
Proof
Let $\mathbf a = \vec {OP}$ and $\mathbf b = \vec {PQ}$.
Then:
- $\vec {OQ} = \mathbf a + \mathbf b$
Let $P'$ and $Q'$ be points on $OP$ and $OQ$ respectively so that:
- $OP' : OP = OQ' : OQ = m$
Then $P'Q'$ is parallel to $PQ$ and $m$ times it in length.
Thus:
- $\vec {P'Q'} = m \mathbf b$
which shows that:
\(\ds m \paren {\mathbf a + \mathbf b}\) | \(=\) | \(\ds \vec {OQ'}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \vec {OP} + \vec {OP'}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m \mathbf a + m \mathbf b\) |
$\blacksquare$
Sources
- 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Addition and Subtraction of Vectors: $5$. Multiplication by a number
- 1927: C.E. Weatherburn: Differential Geometry of Three Dimensions: Volume $\text { I }$ ... (previous) ... (next): Introduction: Vector Notation and Formulae
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 1$.
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Laws of Vector Algebra: $22.5$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): vector space