Unit_6 - UNIT 6 CROSS PRODUCT IN R3 INTRO In this Unit we...

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UNIT 6 CROSS PRODUCT IN R 3 INTRO: In this Unit we will introduce the cross product along with a number of applications, and we will compare the dot product and the cross product. 1. Cross Product In R 2 and R 3 the dot product is defined as < a, b > · < c, d > = ac + bd and < a, b, c > · < d, e, f > = ad + be + cf , respectively. Actually, the dot product exists in all vector spaces R n with vectors < a 1 , a 2 , a 3 , . . . , a n > . However, despite its name, the dot product or inner product, this operation should not be viewed as a multiplication of two vectors, since the ‘product’ produces a scalar and not another vector. Nor should one anticipate being able to multiply two vectors, since the universal characteristic of all vector spaces is that the vectors can be added and multiplied by constants, but not multiplied by each other. There is nevertheless an operation in R 3 (and only in R 3 - not in any other vector space) which looks very much like multiplying two vectors to get another vector. This unusual operation, called the cross product , exists only in R 3 and in no other vector space, and, in fact, is considered by mathematicians to be a skew symmetric tensor product and not really a multiplication of vectors. Still, the cross product of two vectors, ~u and ~v , in R 3 does give another vector ~u × ~v in R 3 defined in the following fashion. DEFINITION: If ~u = < u 1 , u 2 , u 3 > and ~v = < v 1 , v 2 , v 3 > are both vectors in R 3 , then the cross product of ~u and ~v , denoted ~u × ~v , is defined as ~u × ~v = < u 2 v 3 - u 3 v 2 , u 3 v 1 - u 1 v 3 , u 1 v 2 - u 2 v 1 > Since this would be unpleasant, or at least tedious, to memorize, most people remember the cross product as being defined by the following formal determinant: < u 1 , u 2 , u 3 > × < v 1 , v 2 , v 3 > = det ˆ i ˆ j ˆ k u 1 u 2 u 3 v 1 v 2 v 3 Here, the first row of the formal matrix consists of the three coordinate unit vectors in R 3 and the remaining two rows consist of the components of the two vectors. It is important that the second row of the determinant contain the components of the first vector in the cross product, and the third row of the determinant contain the components of the second vector, since we may recall that switching two rows of a determinant will change the sign of the result. This is called a formal matrix and a formal determinant, because a matrix should have numbers as its entries, or at least the same type of object in each entry, but this matrix has vectors in the first row and numbers in the next two rows. In this sense, the determinant is being taken symbolically, simply following the rules for taking determinants of 3 × 3 matrices.
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This idea of formal matrix manipulation occurs elsewhere in mathematics. For example, in advanced calculus and engineering you will run across the curl of a vector field ~ F , which has unit vectors, differentiation symbols and functions inside the determinant: curl ~ F ( x, y, z ) = det ˆ i ˆ j ˆ k x y z F x F y F z = ( y F z - z F y ) ˆ i - ( x F z - z F x ) ˆ j + ( x F y - y F x ) ˆ k
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