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Scalar triple product

Definition. Scalar triple product of vectors (vector product) is a dot product of vector a by the cross product of vectors b and c.

Scalar triple product formula

Scalar triple product of vectors is equal to the determinant of the matrix formed from these vectors.

Scalar triple product of vectors a = {ax; ay; az}, b = {bx; by; bz} and c = {cx; cy; cz} in the Cartesian coordinate system can be calculated using the following formula:

a · [b × c] =  ax  ay  az 
 bx  by  bz 
 cx  cy  cz 

Scalar triple product properties

  • Geometric interpretation.
    Module of scalar triple product of vectors a, b and c is equal to the volume of the parallelepiped formed by these vectors:

    Vparallelepiped = |a · [b × c]|

  • Geometric interpretation.
    The volume of the pyramid formed by three vectors a, b and c is equal to one-sixth of the modulus of the scalar triple product of this vectors:
    Vpyramid1|a · [b × c]|
    6
  • If the mixed product of three non-zero vectors equal to zero, these vectors are coplanar.
  • a · [b × c] = b · (a · c) - c · (a · b)
  • a · [b × c] = b · [c × a] = c · [a × b] = -a · [c × b] = -b · [a × c] = -c · [b × a]
  • a · [b × c] + b · [c × a] + c · [a × b] = 0 - Jacobi identity.

Scalar triple product examples

Example 1. Find the scalar triple product of vectors a = {1; 2; 3}, b = {1; 1; 1}, c = {1; 2; 1}.

Solution:

a · [b × с] =    1     2     3    =
  1     1     1  
  1     2     1  

= 1·1·1 + 1·1·2 + 1·2·3 - 1·1·3 - 1·1·2 - 1·1·2 = 1 + 2 + 6 - 3 - 2 - 2 = 2
pyramid formed by three vectors
Example 2.
Find the volume of a pyramid constructed on vectors a = {1; 2; 3}, b = {1; -1; 1}, c = {2; 0; -1}.

Solution: Calculate scalar triple product of vectors:

a · [b × с] =    1     2     3    =
  1     -1     1  
  2     0     -1  

= 1·(-1)·(-1) + 2·1·2 + 3·1·0 - 3·(-1)·2 - 2·1·(-1) - 1·1·0 =

= 1 + 4 + 0 + 6 + 2 - 0 = 13

Calculate the volume of the pyramid using the following properties:

Vpyramid1|a · [b × c]| = 13 = 21
666

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