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Since addition and subtraction are possible, one might wonder if it is also possible to multiply two vectors. The most obvious idea -- just multiplying corresponding components -- turns out not to be all that useful. However, the following definition has geometric significance that makes it very useful. We will explore this geometry in what follows.
Definition: The dot product of two vectors v = (a,b) and w = (c,d), which we write as <v,w>, is ac + bd.
The dot product of two vectors is a scalar. For this reason the dot product is sometimes called the scalar product. The name "dot product" comes from the fact that another common notation for <v,w> is v.w.
If v and w are the same vector, we find
<v,v> = a2 + b2 = |v|2.
Thus, the length of a vector is the square root of its dot product with itself.
We see that the dot product tells us about the relative direction of the two vectors. If one of the vectors happens to lie on the horizontal or vertical axis of the plane, this direction is relative to our coordinate system.
We can use the vectors i and j and the properties of scalar multiplication and vector addition to write vectors in another way. For example, the vector (4,5) can be written as
(4,5) = 4(1,0) + 5(0,1) = 4i + 5j
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