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/
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