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Unformatted text preview: Dot and Cross Products John E. Gilbert, Heather Van Ligten, and Benni Goetz Dot Product: so far we’ve added vectors, subtracted them, and multiplied by a scalar, but now it’s 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 θ 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 θ , one smaller than π , the other larger than π (unless both equal π ). By convention the smaller one is always chosen, so that ≤ θ ≤ π . Notice that the angle between any two of the unit coordinate vectors i , j , and k is 1 2 π because they are mutually perpendicular . θ v u Definition: The Dot Product u · v of vectors u and v is the scalar defined by u · v =  u  v  cos θ where θ is the angle between u and v . Since cos( 1 2 π ) = 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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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.
 Spring '07
 Sadler
 Multivariable Calculus, Vectors, Scalar, Dot Product

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