The Dot Product Thus far we have discussed the addition, subtraction, and scalar multiplication of vectors. Another operation on vectors is called the dot product. It is important because it can be used to compute the angle between two vectors. There are other properties of the dot product which are covered in a calculus course. We begin by defining the dot product. Dot Product Let 123vvvvijkand 123wwwwijk. The dot product of vand w, denoted by , is a real numbergiven byv w112233v wv wv wv wExample 1:Compute if v w430vijkand 225 wijk. Solution:. 4( 2)( 3)(2)(0)(5)14 v wAs we mentioned above, the dot product can be used to find the angle between two vectors. The relationship is given by the following: Angle Between Two Vectors If vand ware nonzero vectors, then cosv wvwThus, the angle qbetween v and wis given by 1cosv wvwThe vectors vand w are perpendicular if and only if . 0v wTo see why the first formula holds, consider the following diagram. For the sake of simplicity, the vectors shown are only in two dimensions. 1
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