Unit Vectors

Date: 07/02/2002 at 17:41:08
From: J. Smith
Subject: Unit Vectors.


I am trying to solve a math problem that I truly do not understand.  
The problem reads:
"Find the two unit vectors that are collinear with each of the 
following vectors.  (a) vector A = (3, -5)"

That's the first question in this problem, anyway.

I don't understand what this problem is even asking me to do.  Is a 
unit vector only ever equal to 1?  I've done a lot of research in my 
book and on the internet and I still don't understand.  Any help you 
could provide would be GREATLY appreciated.

Thanks ever so much.

Date: 07/02/2002 at 21:04:03
From: Doctor Ian
Subject: Re: Unit Vectors.


Hi,

A unit vector can have any direction, but its length is equal to 
1.  So the following are all unit vectors:

  (0,1)               length^2 = 0^2 + 1^2 = 1

  (1,0)               length^2 = 1^2 + 0^2 = 1

  (1/2, sqrt(3)/2)    length^2 = (1/2)^2 + (sqrt(3)/2)^2 = 1

In fact, if you pick any point on the unit circle (i.e., the 
circle centered at the origin, whose radius is 1), the vector 
from the origin to the point is the unit vector (cos(a),sin(a)), 
where a is the angle from the positive x-axis to the point. 

The easiest way to get a unit vector that is collinear with a 
vector (a,b) is to find the magnitude of the vector, 

  |(a,b)| = sqrt(a^2 + b^2)

and divide both components by that:

  1/|(a,b)| * (a,b) = (a/|(a,b)|, b/|(a,b)|)

Do you see why this will always be collinear with the original 
vector, and why its length will always be equal to 1? 

(Note that the unit vector that points in the _opposite_ 
direction is also collinear.)

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/