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Imagine a Cartesian coordinate system whose origin is associated with two unit vectors, ê and â, in a 2D-space. Now, let 0.5 cm be the unit of length in this coordinate system.

The magnitude of a unit vector is, by definition, 1; Does this mean that the magnitude of one of our unit vectors is 1 unit of length? Or, in other words, does this mean that $\left | â \right |$ = $\left | ê \right |$ = 0.5cm?

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The basis vectors are dimensionless quantities with magnitude 1. You create dimensional vectors to represent positions, velocities, accelerations, forces, etc. by multiplying each basis vector by a dimensional scalar and then adding together. For example,

$$\mathbf{r}=(2\,\text{cm})\hat{\mathbf{e}}+(3\,\text{cm})\hat{\mathbf{a}}$$

In other words, the components of a vector carry its dimensions. That way, the same basis vectors can be used to represent all kinds of different vectorial physical quantities.

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You have defined a unit of length to be equal to $0.5\,\rm cm$ which we can call $1\,\rm ash$.

Suppose you move $5\,\rm ash$ in the $\hat e$ direction.

The displacement is $5\,\rm ash\, \hat e$ which is $2.5\,\rm cm\, \hat e$

This means that $|\hat e| =1$ irrespective of what you are describing and whatever the units that you are using.

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