What's the physical meaning/significance of unit vector?

Question

I know that the unit vectors are vectors whose length is one. But, what is its physical meaning and why it is important? I could not find any questions related to this here and by Googling.

Thanks.

Answer 1

As you know, the unit vector of the vector $\underline{v}$ is given by $$\underline{\hat{v}}=\frac{\underline{v}}{|\underline{v}|}$$

This has many applications in mechanics, but a simple application in geometry is to enable one to write down a vector of specific length $k$, say, in the direction of $\underline{v}$ which is therefore $$k\underline{\hat{v}}$$