Answer:
Ā. (B x C) = B. (AXC) = C.(B x A)
Step-by-step explanation:
Let A = (Ax, Ay, Az), B = (Bx, By, Bz) and C = (Cx, Cy, Cz) be the operation between these three vectors that combines the scalar product with the vector product, it is called the product triple scale or mixed product.
The mixed product is denoted as [A, B, C] and is defined as:
[A, B, C] = A · (B × C)
On the other hand, the triple scalar product is equal to the determinant whose rows are the coordinates of the three (three-dimensional) vectors:
Ax Ay Az
[A, B, C] = A · (B × C) = Bx By Bz
Cx Cy Cz
The triple scalar product is useful when you want to define multiplications between three vectors A = (Ax, Ay, Az), B = (Bx, By, Bz) and C = (Cx, Cy, Cz). The mixed product is denoted as [A, B, C] and is defined as:
[A, B, C] = A · (B × C)
The result is always a scalar quantity.
Now, since A, B and C are three-dimensional vectors, then, | A · (B × C) | is equal to the volume of the parallelepiped defined by A, B and C.
PROPERTIES
A, B and C vectors of R3, then:
The mixed product of vectors A, B and C is cyclic, that is, it does not change if its factors are permuted circularly, but it changes sign if they are transposed:
A · (B × C) = B · (C × A) = C · (A × B)
[A, B, C] = - [B, A, C] = - [A, C, B] = [C, B, A]
In case the mixed product of vectors A, B and C is zero (or null), the three vectors are coplanar.
