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).

ParallelogramLaw.png

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


Examples

$3$ Weights Suspended from Pulleys

Let $3$ bodies with mass be suspended by cords from pulleys like so:

Three-weight-equilibrium.png

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