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Scalar Triple Product

  1. The Scalar Triple Product of 3 3-Dimensional Vectors A, B and C is calculated as follows

    A(B×C)=[ABC]=|AxAyAzBxByBzCxCyCz|=|AxBxCxAyByCyAzBzCz|

    where

    A=Axˆi+Ayˆj+Azˆk
    B=Bxˆi+Byˆj+Bzˆk
    C=Cxˆi+Cyˆj+Czˆk
  2. Following are Properties of Scalar Triple Product
    1. The Scalar Triple Product of 3 Vectors gives the Signed Volume of Parallelopiped Bound by the 3 Vectors
    2. The Value of the Scalar Triple Product Does Not Change if the Dot and the Cross Products are Interchanged as shown below

      A(B×C)=(A×B)C[ABC]=[CAB]
    3. The Value of the Scalar Triple Product Does Not Change on Cyclic Permutation of the Vectors as shown below

      A(B×C)=C(A×B)=B(C×A)

      [ABC]=[CAB]=[BCA]
    4. The Sign of Value of the Scalar Triple Product Changes on Non-Cyclic Permutation of the Vectors as shown below

      A(B×C)=A(C×B)=C(B×A)=B(A×C)

      [ABC]=[ACB]=[CBA]=[BAC]
    5. The Value of Scalar Triple Product is 0 if any 2 Vectors are Parallel or Same, or if the 3 Vectors are Co-Planar (i.e. they lie on the same Plane)
Related Topics
Vector Triple Product,    Scalar Quad Product,    Vector Quad Product,    Introduction to Vector Algebra
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