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The Dot Product

# The Dot Product - a b c we just need to compute the...

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The Dot Product Technically speaking, the dot product is a kind of scalar product. This means that it is an operation that takes two vectors, "multiplies" them together, and produces a scalar. We don't, however, want the dot product of two vectors to produce just any scalar. It would be nice if the product could provide meaningful information about vectors in terms of scalars. What do we mean by "meaningful"? Glad you asked. To begin, let's look for scalar quantities that can characterize a vector. One easy example of this is the length, or magnitude, of a vector v , usually denoted by | v | . Every one of the 2- and 3-dimensional vectors that we have been discussing has length, and length is a scalar quantity. For instance, to find the length of a vector
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Unformatted text preview: ( a , b , c ) , we just need to compute the distance between the origin and the point ( a , b , c ) . (The idea is the same in two dimensions). Our measurement will yield a scalar value of magnitude without direction--not another vector! This type of scalara sounds like the kind of meaningful information the dot product could provide for us. Summary of Dot Product Rules In summary, the rules for the dot products of 2- and 3-dimensional vectors in terms of components are: u · v = u 1 v 1 + u 2 v 2 u · v = u 1 v 1 + u 2 v 2 + u 3 v 3 The rule for vectors given in terms of magnitude and direction (in either 2 or 3 dimensions), where θ denotes the angle between them, is: v · w = | v || w | cos θ...
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