Explaining the Dot Product

Date: 04/05/98 at 16:25:54
From: Paul Waycaster
Subject: dot product

I am currently a high school student teacher teaching trigonometry. We 
are doing a unit on vectors. When inner (dot) product was taught, many 
questions were raised. Everyone understood that when given vector u 
and vector v, the dot product is ||u|| times ||v|| times the cosine of 
the angle between them, but we had a problem when we got the answer. 
Everyone understood that the answer was a scalar, not a vector, but 
there is no graphical representation for what this scalar stands for.  
I have checked numerous sources (every text book I could get my hands 
on, the internet, the math department at the university I am 
attending, and the math department where I am student teaching). I 
have had the students look for an answer on the internet and in the 
library. We did an activity drawing vectors and comparing the dot 
product with the vectors. None of us has been able to find an 
understandable meaning of dot product. We have exhausted our resources 
and hope you can help us. We have done problems involving work and the 
dot product, so we have seen a real world application, but we are 
still confused as to what it really is.

QUESTION:  
Exactly what does the dot product represent? Is there a graphical 
explanation for the resulting scalar? 

Please help us clear the confusion. Thank you.

Date: 04/06/98 at 11:38:41
From: Doctor Anthony
Subject: Re: dot product

You are perhaps thinking of dot products the wrong way around. The dot 
product of the vector (x1, y1, z1) and the vector (x2, y2, z2) is 
written down in a few seconds:

     v1.v2 = x1.x2 + y1.y2 + z1.z2

Now, having this, we can find the angle between v1 and v2, since:

                    v1.v2
     cos(theta) = ---------
                  |v1|.|v2|

where |v1| = sqrt(x1^2 + y1^2 + z1^2), and similarly 
      |v2| = sqrt(x2^2 + y2^2 + z2^2).

By expressing v2 as a unit vector, we can also write down the 
component of v1 in the direction of v2. We can test for two vectors 
being perpendicular, since if they are perpendicular, cos(theta) = 0 
and v1.v2 = 0.

Since it is just as easy to work with vectors in 3 dimensions as in 2 
dimensions, you will find that most 3D geometry is done using vectors, 
and the dot product turns up in just about every problem you can think 
of; for example, finding the distance of a point from a plane or from 
a line, or the shortest distance between two lines in space, or the 
equation of a plane defined by three points. Some of these can also be 
solved using VECTOR products, but that is a more advanced concept.

In short, we don't set out to find the dot product. We set out to find 
angles between vectors, the component of a vector in some direction, 
the distance of a point from a line or plane, the equation of a plane, 
and so on and so on, and we use dot products in getting the answers to 
these questions. In a similar way, you don't multiply two numbers for 
the fun of it. You multiply numbers to answer some question which 
requires the technique of multiplication as an essential aid.

-Doctor Anthony,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/