# How to find unit vector

• Jul 2nd 2012, 11:19 AM
daigo
How to find unit vector
Say for example v = <3,4>

I was taught to divide each component by the magnitude in order to get the unit vector, i.e.

3^2 + 4^2 = ||v||^2
5 = ||v||

So the unit vector of that vector is <3/5,4/5> or 1/5<3,4>

But if I forgot that I had to divide the components by the magnitude, I would not know how to get the unit vector. So I guess I am asking why you divide the components by the magnitude in order to get the unit vector?
• Jul 2nd 2012, 11:47 AM
Plato
Re: How to find unit vector
Quote:

Originally Posted by daigo
But if I forgot that I had to divide the components by the magnitude, I would not know how to get the unit vector. So I guess I am asking why you divide the components by the magnitude in order to get the unit vector?

Unit vector means a vector of length 1. That is why.
• Jul 2nd 2012, 12:38 PM
HallsofIvy
Re: How to find unit vector
If the vector is <a, b> then its length is, as you say, $\sqrt{a^2+ b^2$. If you multiply the entire vector, and so each component, by that, you have
$\left<\frac{a}{\sqrt{a^2+b^2}}, \frac{b}{\sqrt{a^2+ b^2}}\right>$ and its length is
$\sqrt{\left(\frac{a}{\sqrt{a^2+ b^2}}\right)^2+ \left(\frac{b}{\sqrt{a^2+ b^2}}\right)^2}= \sqrt{\frac{a^2}{a^2+ b^2}+ \frac{b^2}{a^2+ b^2}}$
$= \sqrt{\frac{a^2+ b^2}{a^2+ b^2}}= 1$