Complete unit of vectors and their scalar counterparts [closed]

Question

I'm aware that we use m/s as the SI unit for both speed and velocity. My question then is, we use units to define the magnitude of the vector (velocity), why not add units for the direction too?

For a velocity vector in 1-D, why not write the unit as m/s î, for example. (Here î is an arbitrary unit vector and not necessarily the positive X-axis).

Why not express 2-D velocity as m/s (â), where â is the total direction?

I understand that this might seem like a silly question. But, anyways I'm curious about this.

Answer 1

You can write any vector $\vec v$ as a magnitude and a direction as follows: $$\vec v= |\vec v| \left(\frac{\vec v}{|\vec v|}\right)=v (\hat v)$$ where $v$ is the magnitude of $\vec v$ and $\hat v$ is the direction of $\vec v$, also called the unit vector in the direction of $\vec v$.

Now, $v=|\vec v|=\sqrt{\vec v \cdot \vec v}$, so arithmetically $v$ has the same units as $\vec v$. And since $\vec v=v \hat v$ it is clear that $\hat v$ is unitless.

So, directly from the arithmetic it is clear that the direction is unitless and the vector has the same units as the magnitude

Answer 2

The objective of using the SI unit system is to provide a standarised set of useful units that allows people to express relevant magnitudes. From here we can see several problems.

• First, a direction is not a magnitude, so it doesn't really fit in the SI.

• Second, for most experiments, the absolute orientation of the experimental apparatus. You mostly only care about its relative orientation with respect to something (e.g. the laser source, the magnetic field of the earth, the direction of earth's gravity, etc...). So there is no need to introduce an absolute direction.

• Third, if you do care about such a direction, you could always express it as relative to three objects (e.g. several stars), in terms of, for example, the Euler angles. But this would be EXTREMELY dependent on the experimental setup, so making up a standard for this seems useless.