A vector $\vec{B}$ has a magnitude $B$ and and a unit vector $\hat{B}$ in the direction of $B$ then which of the following are correct
1) $\vec{B} .\hat{B} = B$
2) $\hat{B} = \frac {\vec{B}} {B}$
3) $\vec{B}.\vec{B} = B^2$
4) $B = \frac {\vec{B}} {\hat{B}}$
So I was attempting this question and I got my answers as 1), 2), 3), 4)
Now according to my book the answer is 1), 2), 3).
I don't understand why. If $\hat{B} = \frac {\vec{B}} {B}$ is correct then why not $B = \frac {\vec{B}} {\hat{B}}$ because in this we have just interchanged the denominators through cross multiplication right.
Please correct if I am wrong and please justify your answer.
You can't divide vectors. It's just not defined.
As a bit of clarification: you could just as easily say that $\hat{B} = \frac{1}{B} I \vec{B}$, and from that conclude that $\frac{\hat{B}}{\vec{B}} = \frac{1}{B} I$, where I is the unit matrix. Multiplication just doesn't work nicely enough here to allow unambigious division.