Vector Multiplication - Prince George's Community …

1 Vector Simpson, of Physical Sciences and EngineeringPrince George s Community CollegeSeptember 18, 2010 IntroductionWhen dealing with scalars, there is only one definition of Multiplication : you multiply one scalar by anotherscalar, giving a scalar result according to the usual laws of arithmetic:a bDc. But with vectors, therearethreedifferent kinds of Multiplication : one kind gives a scalar result, another gives a Vector result, andanother gives a tensor result. Here we ll summarize the various types of Vector Multiplication and show howto compute each in terms of the rectangular components of the begin, let s represent vectors ascolumn vectors that is,3 1matrices.

2 We ll define the vectorsAandBas the column vectorsAD0@AxAyAz1 AIBD0@BxByBz1A(1)We ll now see how the three types of Vector Multiplication are defined in terms of these column vectors andthe rules of matrix ProductThefirst type of Vector Multiplication is called thedot product. This type of Multiplication (writtenA B)multiplies one Vector by another and gives dot product of two vectorsAandBis the product of their magnitudes times the cosine of the anglebetween them:A BDABcos . In terms of rectangular components, this is equal to the transpose ofcolumn vectorAtimes column vectorB, which gives a1 1matrix ( a scalar):A BDATBD AxAyAz 0@BxByBz1 ADAxBxCAyByCAzBz:(2)The dot product is commutative (A BDB A).

3 Cross ProductThe second type of Vector Multiplication is called thecross product. This type of Multiplication (writtenA B) multiplies one Vector by another and gives a anothervectoras the result of the cross product operation is a Vector whose magnitude isjA BjDABsin ,where isthe angle between the two vectors. The direction ofA Bis perpendicular to the plane containing vectorsAandB, in a right-hand the other two kinds of Multiplication , the cross product is only defined convenient mnemonic forfinding the rectangular components of the cross product is through a matrixdeterminant:A BD ijkAxAyAzBxByBz AzBy/i .AxBz AyBx/k:(3)Another way to represent the components of the cross product is to write the components of vectorAinto a3 3matrix, then multiply that matrix by the column vectorB:A BD0@0 AzAyAz0 Ax AyAx01A0@BxByBz1AD0@AyBz AzByAzBx AxBzAxBy AyBx1A:(4)The cross product isanti-commutative(A BD B A) and non-associative (A.)

4 B B/ C).Direct ProductThe third type of Vector Multiplication is called thedirect product, and is writtenAB. Multiplyingone vectorby another under the direct product gives atensorresult. A tensor is a3 3matrix that is used to representcertain quantities as stress and rectangular components of the direct product may be found by matrix Multiplication : one multipliesthe column vectorAby the transpose ofB, which gives a3 3matrix:ABDABTD0@AxAyAz1A BxByBz D0@AxBxAxByAxBzAyBxAyByAyBzAzBxAzByAzBz1 A:(5)The direct product is non-commutative (AB6 DBA). Vector Product IdentitiesA few Vector product identities are of interest:A B CDA B CDB C ADB C ADC A BDC A B(6)A.

5 B C/ B/(7).A B/ C/ C/(8).A B/ .C C/.B D/ .A D/.B C/(9).A B/ .C B D/C .A B C/D(10)Note that in the Vector triple product A B C, there is no ambiguity in the order of operations: thecross product must be donefirst. (Attempting to do the dot productfirst results in the cross product of ascalar with a Vector , which is not defined.) Eq. (6) says that the order of the dot and cross products maybe interchanged, or the order of the vectors permuted cyclically, without changing the result. The result is ascalar whose absolute value is equal to the volume of a parallelepiped defined by the three products in Eqs. (7) and (8) may be summarized as: The middle Vector times the dot product ofthe two on the ends, minus the dot product of the two vectors straddling the parenthesis times the remainingone.

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