UPSC » UPSC CSE Study Materials » Mathematics » Vector Triple Product Formula

Vector Triple Product Formula

Dot and cross multiplication of three vectors A, B and C may produce meaningful products, called triple products, of the terms (A.B)C, A.(B*C), and A*(B*C). There’s an easy formula for A.(B*C) when the unit vectors i, j, k are used.

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.

 
faq

Frequently asked questions

Get answers to the most common queries related to the UPSC Examination Preparation.

Prove that [a,b,c+d] = [a,bc] + [a,b,d]

Solution: [a,b,c+d] = a.(b*(c+d)) ...Read full

Vector triple product is given by?

Solution: A*(B*C) = (A.C)B – (A.B)C

Is it correct A*(B*C) = (A*B)*C . If not then why?

Solution: No, it’s not correct as associative law for cross products fails. It can be corrected as A*(B*C) ...Read full

What’s the difference between vector triple product and vector product?

Solution: Vector product is numerically enclosed within the parallelogram with sides because the two vectors multipl...Read full

What’s the physical significance of the vector triple product?

Solution: As we all know, the scalar triple product between three vectors represents the volume of a parallelepiped ...Read full