Triple products are two types
1) Scalar triple product
2) Vector triple product
SCALAR TRIPLE PRODUCT
Definition : Consider three vectors , and .
If the product of three vectors is a scalar, then the multiplication is called scalar triple product. It is written as [ ]. It is read as box and defined as
Note : 1) If i, j, k are perpendicular unit vectors, then
[ I j k ] = [ j k I ] = [ k i j ] = 1
[ j i k ] = [ k j I ] = [ I k j ] = - 1
[ i I j ] = [ j j k ] = [ k k I ] = 0
Conceptual theorem
1. If = a1i + a2j + a3k, = b1i + b2j + b3k, = c1i + c2j + c3k, then prove that
Proof: We know that [ ] = . ×
× = i (b2 c3 - b3 c2 ) - j (b1 c3 - b3 c1) + k (b1 c2 - b2 c1)
a. ( b × c ) = (a1i + a2j + a3k) . [ i (b2c3 - b3c2) - j (b1c3 - b3c1) + k (b1c2 - b2 c1) ]
[ a b c] = a1(b2 c3 - b3 c2) - a2(b1 c3 - b3 c1) + a3(b1 c2 - b2 c1) .......... (1)
= a1(b2 c3 - b3 c2) - a2(b1 c3 - b3 c1) + a3 ( b1 c2 - b2 c1) ........(2)
From (1) and (2)