cross_product - The Cross Product When discussing the dot...

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T he C ross P roduct When discussing the dot product, we showed how two vectors can be combined to get a number. Here, we shall see another way of combining vectors, this time resulting in a vector. This operation is called the cross product . The cross product of two vectors v and w is a third vector which is perpendicular to both v and w . Unlike the dot product (which is defined for any dimension of the vector), the cross product is only defined for three-dimensional vectors. The cross product is more easily remembered in terms of a linear algebra result, as the determinant of a 3 μ 3 matrix. Cross Product Let 1 2 3 v v v v i j k and 1 2 3 w w w w i j k . The cross product of v and w , denoted by v μ w , is a vector given by 1 2 3 2 3 3 2 3 1 1 3 1 2 2 1 1 2 3 ( ) ( ) ( v v v v w v w v w v w v w v w w w w i j k v w i ) j k Example 1: Compute v μ w if 4 3 0 v i j k and 2 2 5   w i j k . Verify that v μ w is perpendicular to both v and w . Solution: v μ w = [(–3)(5) – (0)(2)] i + [(0)(–2) – (4)(5)] j + [(4)(2) – (–3)(–2)] k = –15 i – 20 j + 2 k . Notice that and . (4)( 15) ( 3)( 20) (0)(2) 60 60 0 0     v v w 2)( 15) (2)( 20) (5)(2) 30 40 10 0 ( w v w And so, we see that both v and w are perpendicular to v μ w . Similar to the case of the dot product, there is a nice geometrical interpretation of the cross product, this time in terms of areas of parallelograms. To that end, we make the following claim. 1
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