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:

$(a, b, c) =\begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix}$

References