Let a, b, c be any three vectors, then the expression a*(b*c) could be a vector and is named a vector triple product. Consider the expression a*(b*c) which itself maybe a vector, since it is a cross product of two vectors a and (b*c). Now a*(b*c) could be a vector perpendicular to the plane containing a and (b*c) but b*c could be a vector perpendicular to the plane containing b and c, therefore a*(b*c) could be a vector that lies within the plane of b and c and perpendicular to a.
Hence we will express a*(b*c) in terms of b and c i.e. a*(b*c) = xb + yc , where x & y are scalars.
The Value Of Vector Triple Product
Suppose A, B, and C are vectors and m may be a scalar. Then the following laws hold:
1) Generally, (A.B)C ≠ A(B.C).
2) A.(B*C) = B.(C*A) = C.(A*B) = volume of a parallelepiped having A, B, and C as edges,
or the negative of this volume, according to as A, B, and C do or don’t form a
right-handed system.
3) Normally, A*(B*C) ≠ (A*B)*C
(Associative Law for Cross Products Fails)
4) A*(B*C) = (A.C)B – (A.B)C
(A*B)*C = (A.C)B – (B.C)A
Vector Triple Cross Product Formula
A*(B*C) = (A.C)B – (A.B)C
and
(A*B)*C = (A.C)B – (B.C)A
In general, A*(B*C) ≠ (A*B)*C
Vector Triple Product Formula Proof
Let product be a*(b*c)
Product can be written as the linear combination of vectors a and b.
Hence, the product are often be written as (a*b)*c = xa + yb
So we’ll proceed as,
c.(a*b) * c = c.(xa + yb)
x.(c.a) + y.(c.b)
x.(a.c) + y.(b.c) = 0
Therefore,
x/(b.c) = -y/(a.c) = λ
So we get,
x = λ(b.c) and y = λ(a.c)
Since we’ve (a*b)*c = xa + yb ——— (1)
So let’s substitute values of x and y in equation (1)
So we are going to get
(a*b)*c = (λb.c)a + (-λa.c)b = (λb.c)a – (λa.c)b
Product is for every value of a, b and c and also the reason is each of them has an identity.
So, put a = i, b= j and c = i
(i*j)* = (λj.i)i – (λi.i)j
j = -λj
λ = -1
Therefore,
(a*b)*c = (a.c)b – (b.c)a
Applications Of Vector Triple Product
1) It’s a vector product.
2) It’s used to find the unit vector coplanar with a and b and perpendicular to c.
Conclusion
In conclusion, if a, b, c be any three vectors, then the expression a*(b*c) could be a vector and is named a vector triple product. We are able to express a*(b*c) in terms of b and c i.e. a*(b*c) = xb + yc , where x & y are scalars. There’s an easy formula for A.(B*C) when the unit vectors i, j, k are used. It’s very useful to find the coplanarity of the vectors.