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ME 6590 Multibody Dynamics Representation of Vector Operations as Matrix Operations Dot (or Scalar) Product Given two vectors a a1e1 a2e2 a3e3 b b1e1 b2e2 b3e3 then the dot (or scalar) product of the two vectors is defined to be 3 a b aibi . i 1 This product can be represented as the following matrix product a b {a}T {b} where {a} indicates a 3 1 column vector whose elements are the components of the vector a , and {a}T indicates a 1 3 row vector that is the transpose of {a} . Recall that for any two vectors a and b , a b b a . That is, the dot product is commutative. The matrix equivalent of this statement is {a}T {b} {b}T {a} . Cross (or Vector) Product Given two vectors a a1e1 a2e2 a3e3 b b1e1 b2e2 b3e3 then the cross (or vector) product of the two vectors is defined to be a b (a2b3 a3b2 )e1 (a3b1 a1b3 )e2 (a1b2 a2b1 )e3 . This product can be represented as the following matrix product 1/2 0 a b [a]{b} a3 a2 a3 0 a1 a2 b1 a1 b2 0 b3 where [a] is a skew-symmetric matrix (defined above) containing the components of a . Recall that for any two vectors a and b , a b b a . The matrix equivalent of this statement is [a]{b} [b ]{a} . 2/2