Subtraction of Vectors

Vector subtraction is not a new operation — it is addition of a reversed vector. To find $\vec A - \vec B$, first construct $-\vec B$ (same magnitude as $\vec B$, opposite direction), then apply the ordinary triangle law of addition to $\vec A$ and $-\vec B$:

If $\theta$ is the angle between $\vec A$ and $\vec B$, then the angle between $\vec A$ and $-\vec B$ is $180° - \theta$. Substituting $180°-\theta$ in place of $\theta$ in the addition formula (and using $\cos(180°-\theta) = -\cos\theta$, $\sin(180°-\theta)=\sin\theta$) gives:

where $\alpha_2$ is the angle the difference vector makes with $\vec A$. Notice the only change from the addition formulas is the sign in front of the $2AB\cos\theta$ (and $B\cos\theta$) term — subtraction is 'addition with the angle supplemented'. …