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### Explaining the Dot Product

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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?

```

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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/
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Associated Topics:
College Linear Algebra
College Trigonometry
High School Linear Algebra
High School Trigonometry

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