Addition of Vectors

Because vectors carry direction, $\vec A + \vec B$ cannot be found by ordinary addition of the numbers $A$ and $B$ — it needs a genuinely geometric (or, equivalently, component-wise) rule. The triangle law of addition gives that rule for two vectors inclined to each other at some angle:

Draw $\vec A$ and $\vec B$ as two sides of a triangle, taken in the same order — i.e. draw $\vec A$ first, then draw $\vec B$ starting from the head (tip) of $\vec A$. The resultant $\vec R = \vec A + \vec B$ is then the third side of the triangle, drawn from the tail of $\vec A$ directly to the head of $\vec B$.

Magnitude of the resultant. Let $\theta$ be the angle between $\vec A$ and $\vec B$. Extending the side representing $\vec A$ and dropping a perpendicular from the head of $\vec B$ onto that extension creates a right triangle from which $AN = B\cos\theta$ and $BN = B\sin\theta$. Applying Pythagoras to this right triangle:

Direction of the resultant. If $\vec R$ makes angle $\alpha$ with $\vec A$, then from the same right triangle, …