Clarification On The Unit Vector

Question

I am reading the text on Mechanics by Kleppner and Kolenkow. In the part about vectors, it states that a unit vector is as follows: $$\hat{A} = \frac{A}{|A|}$$ I'm not sure what this is saying. Does it mean a unit vector $ \hat A $ is equal to vector $A$ divided by it's magnitude?

Answer 1

Intuitively, what this means is that we take the vector (magnitude and direction) and divide by its magnitude $\lvert A\rvert$. This means that the magnitude of the resultant vector $$ \hat{A}=\frac{A}{\lvert A\rvert}$$ is $1$. That is, $\hat{A}$ is the vector which tells us the direction in which $A$ points. (A vector of length $1$ is called a unit vector.)

Another nice way to see this is that we can write the vector $A=\lvert A\rvert \hat{A}$ where $\hat{A}$ determines the direction in which $A$ points, and $\lvert A\rvert$ scales the unit vector to the appropriate length.

Answer 2

Yes, that is exactly right, because this means that the resulting vector $\hat{A}$ has unit length, i.e., a length of $1$, but has the same direction as the original vector $A$.