Multiplication of velocity vectors (times a scalar)

You may have noticed that using the Cartesian representation to represent displacement vectors (i.e. \(\vec{m}_1 = \langle 1, 2, 3\rangle \)) is a little inconvenient. One shortcoming they have is that it is not immediately clear how far one moves when undergoing the displacement \(\vec{m}_1\). A quick calculation makes it clear that we move a distance \(l_1 = \sqrt{1^2+2^2+3^2} = \sqrt{14}\), suggesting that our displacement (and all its vector representations) have the length \(l_1\). But now we have the distance travelled in the length \(l_1\) as well as in the direction suggested by \( \langle 1, 2, 3\rangle \).

A common way to ease notation is to explicitly break vectors up into a scalar length and a direction vector such that all direction vectors have the same length, 1. We say that these are unit vectors (because they have a length of one unit), in the direction that they point. For example, the displacement \(\vec{m}_1\) given by \(\vec{m}_1 = \langle 1, 2, 3\rangle \) has length \(l_1 = \sqrt{14} \approx 3.74\) in the direction \( \vec{e}_1 = \langle \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}\rangle \).

We can then say that \(\vec{m}_1 = l_1\vec{e}_1\) and it is clear what we mean. A movement of length \(l_1\) in the direction \(e_1\). Here \(l_1\) is a number, and is inherently independent of the coordinate system. \(\vec{e}_1\) too is independent of coordinates, but its specific representation will depend on the system we use (cylindrical, polar, Cartesian, et cetera).

If we want to repeat the displacement \(\vec{m}_1\) two times, we can make the new displacement \(\vec{m}_2 = 2 \vec{m}_1 = 2l_1\vec{e}_1\). This repetition is not limited to integer numbers, we can have \(\vec{m}_\frac{3}{7} = \frac{3}{7} \vec{m}_1\), \(\vec{m}_\pi = \pi \vec{m}_1\), or in general \(\vec{m}_t = t\vec{m}_1 = tl_1\vec{e}_1\) that represents a continuous motion in time.

As we explore more topics in mechanics, electrodynamics, relativity, and beyond, the extension and contraction of vectors along their direction will be a fundamental tool.