Finding Two Orthogonal Vectors in R3

Date: 12/15/2003 at 21:03:14
From: Emerson
Subject: Find two non zero vectors in R3 .......... 

I'm supposed to find two non-zero vectors in R3 that are orthogonal,
but I'm pretty confused about the whole idea.  Can you explain it to me?

Date: 12/25/2003 at 20:15:46
From: Doctor Jordan
Subject: Re: Find two non zero vectors in R3 .......... 

Dear Emerson,

R3 is three dimensional space; that is, it contains vectors with three 
entries.  Therefore, [1] is not in R3, while [1] is in R3.
                     [1]                     [1]
                     [1]                     [1]
                     [1]

(Note that there are different R3s; in R4 there are infinitely many 
different R3s.  But any three dimensional vector can be expressed in 
an R3, so a three dimensional vector is said to be in R3.)

A non-zero vector is a vector that has at least one non-zero entry. 
As such, [0] is a zero vector while [1] is a non-zero vector.
         [0]                        [0]
         [0]                        [0]

To determine if two vectors are orthogonal (perpendicular), we take 
the dot product of them (the dot product is the sum of the products 
of corresponding entries, from 1 to k, of k dimensional vectors).  If 
and only if the dot product of the vectors is equal to 0, the vectors 
are orthogonal.

For example, let's see if the two vectors below are orthogonal.  We
multiply each corresponding entry, then add up the results to see if
we get zero:

  [2]   [3]    2 * 3 = 6
  [3]   [4]    3 * 4 = 12
  [-2]  [9]   -2 * 9 = -18

  (6) + (12) + (-18) = 0, so these two vectors are orthogonal

Has this helped you?  If you have questions on any of this, or other
questions, please write me back.  Good luck!

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