Adding Vectors

Adding vectors is a drawing exercise before it is an algebra exercise. Get the picture right and the formula falls out of a single right triangle.

Two ways to draw the same sum#

To add $\vec a$ and $\vec b$ with the triangle rule, place the tail of $\vec b$ on the head of $\vec a$. The arrow from the tail of $\vec a$ to the head of $\vec b$ is the sum.

The parallelogram rule says the same thing differently. Draw both vectors from a common tail, complete the parallelogram they span, and the diagonal from the shared tail is the sum. It is the triangle rule seen twice: the far sides of the parallelogram are just copies of $\vec a$ and $\vec b$, slid into place.

How long is the sum?#

Let the magnitudes be $a$ and $b$, with angle $\theta$ between the vectors when drawn tail to tail. In the head-to-tail picture, extend the line of $\vec a$ and drop a perpendicular from the far corner D to meet it at E.

Side BE is the part of $\vec b$ that runs along $\vec a$: $b\cos\theta$. Side DE is the part that sticks out sideways: $b\sin\theta$. Pythagoras on the right triangle ADE gives

and the last bracket is exactly 1. So:

The same triangle also tells us the direction. If $\alpha$ is the angle the sum makes with $\vec a$, then opposite over adjacent in triangle ADE gives

Check the extreme cases in the graph below. At $\theta = 0$ the cosine is 1 and the sum is simply $a + b$. At $\theta = 180^\circ$ the cosine is −1 and you get $|a - b|$. At $90^\circ$ the middle term disappears and you are back to Pythagoras.