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Dot Product

Dot Product - Dot Product The next topic for discussion is...

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Dot Product The next topic for discussion is that of the dot product. Let’s jump right into the definition of the dot product. Given the two vectors and the dot product is, (1) Sometimes the dot product is called the scalar product . The dot product is also an example of an inner product and so on occasion you may hear it called an inner product. Example 1 Compute the dot product for each of the following. (a) (b) Solution Not much to do with these other than use the formula. (a) (b) Here are some properties of the dot product. Properties

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The proofs of these properties are mostly “computational” proofs and so we’re only going to do a couple of them and leave the rest to you to prove. Proof of We’ll start with the three vectors, , and and yes we did mean for these to each have n components. The theorem works for general vectors so we may as well do the proof for general vectors. Now, as noted above this is pretty much just a “computational” proof. What that means is that we’ll compute the left side and then do some basic arithmetic on the result to show that we can make the left side look like the right side. Here is the work.
Proof of : If then This is a pretty simple proof. Let’s start with and compute the dot product. Now, since we know for all i then the only way for this sum to be zero is to in fact have . This in turn however means that we must have and so we must have had .

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Dot Product - Dot Product The next topic for discussion is...

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