Unit vectors give us a short-hand way to describe vectors.

A unit vector is simply a vector with a magnitude of one that points along one of the coordinate axes (x, y, or z).

For example, a displacement vector with a length of one meter obviously has a length of one.  A velocity vector with a magnitude of one m/s also has a “length” of one.  The better word is magnitude, but sometimes people refer to it as a length.

Unit vectors x hat and y hatWe denote unit vectors using the letter of the coordinate direction, capped with a hat, i.e. \(\color{black}{\hat{x}, \hat{y}, \mathrm{or} \hat{z}}\).  We pronounce these symbols as x-hat,  y-hat, or z-hat.

Unit vector x hat and a vector 2 m x hatIf we have a vector that is 2 m long that points in the x-direction, we write it symbolically as 2 m  \(\color{black}{\hat{x}}\).  Sometimes for clarity we put a set of parentheses around the length of the vector.   We could also write this vector as (2 m) \(\color{black}{\hat{x}}\).