dot_product

dot_product - The Dot Product Thus far we have discussed...

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T he D ot P roduct 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 123 vv v  vi j k and ww w  wi j k . The dot product of v and w , denoted by , is a real number given by vw 11 2 2 33 Example 1: Compute if 430  j k and 225   j k . Solution: . 4( 2) ( 3)(2) (0)(5) 14 As 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 v and w are nonzero vectors, then cos v w Thus, the angle q between v and w is given by 1 cos    The vectors v and w are perpendicular if and only if . 0 To 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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x y v v 1 , 2 w w 1 , 2 v - w q Figure 1: Visualizing the Law of Cosines We have that 12 vv  vi j , 1 ww 2 wi j and 11 2 2 () ( vw )   i j . By the Law of Cosines, we have that 22 2 2c o s  v w . First off, observe that 2 22 2 1 1 2 2 2 2 1 2 1 2 ( ) ( ) 2 ( 2 w w w w v w v w  v w 2 ) Thus, our equation becomes 222 c o s  v w v w v w Subtracting || v || 2 and || w || 2 from both sides, we have: c o s v w Finally, dividing both sides by -2, we get the desired formula.
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This note was uploaded on 09/07/2011 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

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dot_product - The Dot Product Thus far we have discussed...

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