VECTOR PRODUCT PROOF

This article will introduce you to the basics of vector algebra. Here we will see what can be the various combinations of the triple product. Readers will acquire the knowledge of the basic theorems like theorem for the plane, and theorem for the plane. Vector is a subject of importance in the subjects in Physics too. The geometrical meaning of the vector triple product formula will be conceptually clear to you if you know what the theorem for plane states. It says that if a vector is coplanar with the two vectors, it can be represented as the linear combination of those two vectors    

POSSIBLE TRIPLE PRODUCT SCENARIOS

Let’s go through the following triple products of the vectors a,b and c and try to figure out their meaning.

1. (a.b).c

2. (a.b)*c

3. (a*b).c

4. (a*b)*c

Out of all the products of 3 vectors, only 3 and 4 are defined and have some meaning. Readers are advised to stop and think for a while about why 1 and 2 don’t have any meaning.

 a.b is the dot product of two vectors. Thus it is obvious that it is a scalar quantity. Thus its dot product is not defined with any vector quantity. Now, what about 2. Now again a.b is a scalar quantity so we can’t take its cross product with some other vector.

But in 3 a*b is a vector so its dot product with another vector is defined. (a*b).c is called scalar triple product or box product and can also be written as [abc]. It represents the volume of the parallelepiped with the adjacent vectors a,b and c. And in 4 a*b is a vector and thus we can take its cross product with some other vector. It is called the vector triple product. Vector triple product formula can be written as;

(a*b).c= b(a.b)-a(b.c)

We will go through the proof of the product of 3 vectors defined as the vector triple product

MEANING OF VECTOR TRIPLE PRODUCT

(a*b)*c will be calculated by first taking the cross product of a and b then we will take the cross product of that vector with the vector c. But we also have a simpler vector triple product formula.

(a*b).c= b(a.b)-a(b.c) this formula is also derived from the geometric meaning of the vector triple product. Let’s try to visualise the geometrical meaning of the vector triple product.

a*b is the vector perpendicular to both a and b. This means the cross product of the two vectors is perpendicular to the plane containing those two vectors. Now say the plane containing the two vectors a and b as P. If we again take the cross product of the vector normal to this plane, with some other vector say c, it will be perpendicular to the plane of the normal and the other vector. Thus the resultant vector triple product will be the vector which will be coplanar with the vectors a and b and perpendicular to the vector c.

Now let’s try to confirm if both ways of calculating the vector triple product hold good or not. We will now verify the vector triple product formula through vector triple product examples.

a= 3i-j+2k      b=2i+j-k    c=i-2j+2k

Now try to calculate the vector triple product a*(b*c) using the basic manner..

b*c= -5j-5k( say d)

Now a*d calculated and found to be

 a*d=15i+15j-15k

Now calculate using the vector triple product formula

a*(b*c)=(a.c)b-(a.b)c

=9b-3c

=15i+15j-15k

Thus it is verified that the vector triple product formula holds good.

VECTOR TRIPLE PRODUCT FORMULA PROOF

Vector triple product formula proof comes from a very basic concept known as a theorem for space. It states that if a vector is coplanar with the two vectors, it can be represented as the linear combination of those two vectors. And now we have a clear explanation that the vector triple product a*(b*c) is perpendicular to a and coplanar with b and c. Thus we can say

a*(b*c)=pb-qc (using the theorem for plane)

Now take the dot product with a both sides

a*[a*(b*c)]=pb.a+qc.a

0=pb.a+qc.a

It gives

p/c.a=-q/b.a=λ

a*(b*c)=pb-qc

 =(λc.a)b-λ(b.a)c

Since this identity holds for every selection of the three vectors

Let us assume a=i,b=j and c=i

a*(b*c)=(λc.a)b-λ(b.a)c

j=λj

Thus λ=1

This way we have proved the vector triple product formula.

CONCLUSION

So a long we have seen what is the geometrical meaning of the vector triple product. The vector triple product formula can be easily depicted by using the theorem for a plane in which we can depict the vector triple product as the linear combination of two other vectors which are coplanar with it. You can also verify the vector triple product formula through the vector triple product examples. It is also obvious now how various other combinations of the product of 3 vectors are not valid and are meaningless.