Adding velocity vectors

We've seen that we can repeat a displacement \(n\) times by multiplying its length by the scalar \(n\), i.e.

\[\vec{m}_1 \rightarrow \vec{m}_n = n \vec{m}_1\]

Another way to think of this repetition is as the addition of \(n\) copies of \(\vec{m}_1\):

\[\vec{m}_n = \overbrace{\vec{m}_1 + \ldots + \vec{m}_1}^{n\text{ times}}\]

For example, consider the case of \(\vec{m}_2 = 2 \vec{m}_1\):

What if we have two distinct displacement vectors \(\vec{m}_1\), and \(\vec{m}_2\) that we wish to compose? How can we combine the two so that we undertake the displacement one \(\vec{m}_1\), followed by displacement two \(\vec{m}_2\)?

Let's draw them out:

We can see that the way to add displacement vectors is to line them up tip to tail. The resultant displacement is the vector that spans from the tail of the first to the tip of the second. Notice that the resultant vector is independent of the order. We can do either first, and we always end up with the same sum.

As always, the result of a vector operation is independent of the representation, and so is our sum. However, in the Cartesian coordinate system, the sum of two vectors can be calculate in a particularly convenient way. Suppose we have two displacement vectors \(\vec{m}_1 = \langle 1, 2, 3\rangle\) and \(\vec{m}_2 = \langle 2, -3, 7\rangle\). If we follow our arrow drawing carefully, we see that the components of the sum is given simply by adding the components of \(\vec{m}_1\) and \(\vec{m}_2\) in pairwise fashion:

\[\begin{align} \vec{m}_1 + \vec{m}_1 & = \langle x_1, y_1, z_1\rangle + \langle x_2, y_2, z_2\rangle \\ &= \langle x_1 + x_2, y_1 + y_2, z_1 + z_2 \rangle \end{align}\]

By composing displacement vectors, we can move up, down, to the side, diagonally, or backwards in a series of steps. It is even possible to go nowhere through a series of intermediate steps, which is what happens when we walk in a circle:

Breaking complex trajectories into a composition of vector displacements can give elegant arguments and explanations for some of the most complex rules of the universe, as Feynman showed in his book on quantum electrodynamics, QED.