Prove the triple scalar product? Ā. (B x C) = B. (AXC) = C.(B x A)

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  CzThe 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.