Parallelogram Law for Vector Subtraction

Theorem

Let $\mathbf u$ and $\mathbf v$ be vectors.

Consider a parallelogram, two of whose adjacent sides represent $\mathbf y$ and $\mathbf v$ (in magnitude and direction).

Then the diagonal of the parallelogram connecting the terminal points of $\mathbf u$ and $\mathbf v$ represents the magnitude and direction of $\mathbf u - \mathbf v$, the difference of $\mathbf u$ and $\mathbf v$.


Proof

We can construct a parallelogram as follows:

and the construction is apparent.


Sources

  • 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach
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