Math.by - Mixed product of vectors

Mixed product of vectors.

Enter coordinates of vectors:

a = ( , , )
b = ( , , )
c = ( , , )

(a,b,c) =

Theory

Mixed product of three vectors a, b, c is the number equal to the vector product a x b, multiplied by the vector c.

(a, b, c) = (a x b, c)

The absolute value of the mixed product (a, b, c) is equal to the volume of the parallelepiped formed by vectors a, b, c (the geometric meaning of the mixed product).

Properties of vectors mixed product:

  • permutation of any two of the factors changes the sign of the product:

    (a, b, c) = - (b, a, c) = (b, c, a) = - (c, b, a) = (c, a, b) = - (a, c, b)

  • if the mixed product is zero ((a, b, c) = 0), then vectors a, b, c is complanar - A necessary and sufficient condition for coplanarity of vectors
  • if the vectors are defined by their coordinates:

    a = (x1, y1, z1), b = (x2, y2, z2), c = (x3, y3, z3)

    mixed product of three vectors is defined by the following formula:

    the formula of mixed product




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