Parallelogram Law
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Theorem
Let $\mathbf a$ and $\mathbf b$ be vector quantities.
Consider a parallelogram, two of whose adjacent sides represent $\mathbf a$ and $\mathbf b$ (in magnitude and direction).
Then the diagonal of the parallelogram through that common point represents the magnitude and direction of $\mathbf a + \mathbf b$, the sum of $\mathbf a$ and $\mathbf b$.
Proof
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Examples
$3$ Weights Suspended from Pulleys
Let $3$ bodies with mass be suspended by cords from pulleys like so:
The bodies will arrange themselves into equilibrium when the vector corresponding to the weight $\mathbf F_3$ of the middle body is equal and opposite the vector corresponding to the vector sum of the weights $\mathbf F_1$ and $\mathbf F_2$ according to the Parallelogram Law.
Sources
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- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $3$. Addition and Subtraction of Vectors
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 1$.
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- 1961: I.M. Gel'fand: Lectures on Linear Algebra (2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules
- 1966: Isaac Asimov: Understanding Physics ... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: Combination of Forces
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- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Vector Interpretation of Complex Numbers
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): parallelogram law: 2.
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): parallelogram rule or parallelogram law
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): parallelogram law
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): parallelogram law
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): addition (of complex numbers)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): parallelogram law