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### 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.
"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/
```
Associated Topics:
College Linear Algebra
High School Linear Algebra

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