Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
3
Academic year
2023
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91
Dot product geometric formula ● The dot product in vectors is a way to calculate the angle between two vectors. It can also be used as a scalar value to determine the magnitude of a vector, or thedistance between two points. ● The dot product of two vectors, A and B, is calculated by multiplying each component of each vector together. The result is a scalar quantity (a singlenumber). If you don't quite remember which of these things it is, a good thing to do is to try and seeif you can identify it by considering what kind of object it is. What is an effective way to help us remember something like this? Memorizing the formulamight feel arbitrary and superficial if we don't also understand why it works. You can compare it to that first formula. It is nice to work out examples for a few simple problems that illustrate the generalconcepts. We can then test my knowledge of dot products by looking at the simpleexamples. People can think of an example in which it would be simpler to figure out what is going onhere. 0 and 90… Once again, to help us remember, we look at some simple examples like these. The suggestion was to look at a vector in that direction and another vector in that direction. Probably this could be the vector (1 , 0) and this one could be the vector (0 , 1).
The angle between them is obvious, it is a right angle. Thus it is pi over 2. Now let’s test.v.w = (1 , 0), dotted with (0 , 1). If we plug in the formula for the product of two numbers, xy = x * y, then we get 0. This isbecause 0 times anything is 0; therefore, we can add them up. And that only matches one of these because the sine of pi over 2, and the tangent of piover 2 are not equal to 0. It's the cosine of pi over 2 that's 0. So that's equal to the norm ofv times the norm of w, times the cosine of pi over 2. This can be simplified to 0, since thecosine of pi over 2 is 0. Yes, a good example. Now consider a slightly more general example, where it turns out to be still pretty clear andelegant to check that this equals that. I wanted to use that example to help demonstratehow the formula works. A general example is that the vector v goes in the x direction. Here, the letter v stands for any number, but zero after comma makes it special. And w isjust any other digit. The vector w is (w1 , w2). Theta is the angle between the vectors. If you take the dot product of v and w, you get v1 times w1. This is because v2 is zero. Thus, I have v1 times w1 plus 0 times w2. Now, if I try to relate v1 and w1 back to thelength of v and the length of w, it turns out that v1 is equal to the length of v.
The length of w1 is the length of w times the cosine of theta. The length of w is equal to the length of v times cosine theta. The dot product, v · w, is equal to v1 times w1. And so it's the norm of v times the normalw times the cosine of v.
Dot Product Geometric Formula
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