# Parallelogram Law

## 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 $\mathbf b$.

## 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.