Resultant in Terms of Dot Product

Theorem
Let $\mathbf a$ and $\mathbf b$ be vector quantities.

Let their resultant be $\mathbf v$:
 * $\mathbf v = \mathbf a + \mathbf b$

Then:
 * $\mathbf v^2 = \mathbf a^2 + 2 \mathbf a \cdot \mathbf b + \mathbf b^2$

where:
 * $\mathbf v^2$ denotes the square of $\mathbf v$
 * $\mathbf a \cdot \mathbf b$ denotes the dot product of $\mathbf a$ and $\mathbf b$

Also presented as
This result can also be presented as:


 * $v^2 = a^2 + 2 a b \cos \theta + b^2$

where:
 * $v$, $a$ and $b$ are the magnitudes of $\mathbf v$, $\mathbf a$ and $\mathbf b$ respectively
 * $\theta$ is the angle between the directions of $\mathbf a$ and $\mathbf b$.

Also see

 * Law of Cosines