cross_product

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 123 vv v  vi j k and ww w  wi j k . The cross product of v and w , denoted by v μ w , is a vector given by 2 3 3 2 3 1 1 3 1 2 2 1 () ( ) ( vvv v w v w v w v w v w v w www  ij k vw i ) j k Example 1: Compute v μ w if 430  j k and 225   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  vvw 2)( 15) (2)( 20) (5)(2) 30 40 10 0  ( wvw 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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Cross Product (Geometrical Interpretation) Suppose q is the angle between the vectors v and w (so 0 § q § p ). Then we have that Area of parallelogram sin determined by and  vw To see why the first equality is true, we work through a bit of algebra.
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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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cross_product - The Cross Product When discussing the dot...

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