# Vectors , magnitude and unit vectors.

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.

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.
• 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. May 14, 2019 at 18:12