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Unformatted text preview: Dot and Cross Products John E. Gilbert, Heather Van Ligten, and Benni Goetz Dot Product: so far weve added vectors, subtracted them, and multiplied by a scalar, but now its time to multiply two vectors. There are two different products, one producing a scalar, the other a vector. Both, however, have important applications to geometry as well as physics and engineering. The angle between vectors u and v is the angle shown in the figure to the right by first arranging them so that they have the same tail. Notice that there are really two choices of , one smaller than , the other larger than (unless both equal ). By convention the smaller one is always chosen, so that . Notice that the angle between any two of the unit coordinate vectors i , j , and k is 1 2 because they are mutually perpendicular . v u Definition: The Dot Product u v of vectors u and v is the scalar defined by u v = | u || v | cos where is the angle between u and v . Since cos( 1 2 ) = 0 , vectors u , v are perpendicular when u v = 0 and u , v = 0 . A number of properties follow immediately from this definition and the perpendicular vectors i , j , k : 1. u v = v u , u u = | u | 2 , 2. u ( v + w ) = u v + u w , 3. i j = 0 , j k = 0 , k i = 0 . Properties: The previously listed properties provide a convenient algebraic way of computing the dot product of vectors u = u 1 , u 2 , u 3 = u 1 i + u 2 j + u 3 k , v = v 1 , v 2 , v 3 = v 1 i + v 2 j + v 3 k . For by expanding using also Properties 1, 2, and 3, we get u v = ( u 1 i + u 2 j + u 3 k ) ( v 1 i + v 2 j + v 3 k ) = u 1 v 1 + u 2 v 2 + u 3 v 3 . Example 1: determine the dot product of the vectors a =- 3 , 2 ,- 3 , b = 2 , 1 ,- 3 ....
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