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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 (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.
The Pythagorean Theorem tells us that the length of a vector (a, b, c) is given by . This gives us a clue as to how we can define the dot product. For instance, if we want the dot product of a vector v = (v1, v2, v3) with itself (v·v) to give us information about the length of v, it makes sense to demand that it look like:
|v·v = v1v1 + v2v2 + v3v3|
Hence, the dot product of a vector with itself gives the vector's magnitude squared.
Ok, that's what we wanted, but now a new question reigns: what is the dot product between two different vectors? The important thing to remember is that whatever we define the general rule to be, it must reduce to whenever we plug in two identical vectors. In fact, @@Equation @@ has already been written suggestively to indicate that the general rule for the dot product between two vectors u = (u1, u2, u3) and v = (v1, v2, v3) might be:
|u·v = u1v1 + u2v2 + u3v3|
This equation is exactly the right formula for the dot product of two 3-dimensional vectors. (Note that the quantity obtained on the right is a scalar, even though we can no longer say it represents the length of either vector.) For 2-dimensional vectors, u = (u1, u2) and v = (v1, v2), we have:
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