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
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 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} .
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