Definition: Consider the three vectors ,
and
. If the product of three vectors is a
vector then the multiplication is called vector triple product.
It is denoted by ( ×
) ×
(or)
× (
×
) and is defined as
CONCEPTUAL THEOREM
1. Prove that ( ×
) ×
= (
.
)
- (
.
)
Proof: Let = a1i
= b1i + b2 j
= c1i + c2 j + c3k
( ×
) ×
= i (0 - a1b2c2) - j (0 - a1b2c1) + k (0 - 0)
( ×
) ×
= -a1b2c2 i + a1b2c1 j → (1)
.
= c1a1
( .
)
= c1a1(b1i + b2j)
( .
)
= c1a1b1 i + c1a1b2 j → (2)
.
= c1b1 + c2b2
( .
)
= (c1b1 + c2b2) a1i
(.
)
= c1b1a1 i + c2b2a1 i →(3) (2) - (3)