For any two vectors u and v show that :
( Vector u.vector v )^2 = [(vector u× vector v)×vector v ].vector u = u^2v^2
To prove the given equation, we'll use the properties of vector dot product and cross product.
First, let's start with the left-hand side of the equation:
(Vector u · Vector v)^2
Using the definition of the dot product, we have:
(u1v1 + u2v2 + u3v3)^2
Expanding the squared term, we get:
(u1v1 + u2v2 + u3v3) * (u1v1 + u2v2 + u3v3)
Now, let's multiply each term and simplify:
= (u1^2v1^2 + u2^2v2^2 + u3^2v3^2) + (u1u2v1v2 + u1u3v1v3 + u2u3v2v3) + (u1u2v1v2 + u1u3v1v3 + u2u3v2v3) + (u1u2v1v2 + u1u3v1v3 + u2u3v2v3)
Notice that the cross terms are repeated three times, so we can simplify it further:
= u1^2v1^2 + u2^2v2^2 + u3^2v3^2 + 3(u1u2v1v2 + u1u3v1v3 + u2u3v2v3)
Now, let's move to the right-hand side of the equation:
[(Vector u × Vector v) × Vector v] · Vector u
Using the definition of the cross product, we have:
[(u2v3 - u3v2)i + (u3v1 - u1v3)j + (u1v2 - u2v1)k] × Vector v
Expanding the cross product, we get:
[(u2v3 - u3v2)v1 - (u3v1 - u1v3)v2 + (u1v2 - u2v1)v3] · Vector u
Now, let's multiply each term and simplify:
= (u2v3v1 - u3v2v1 - u3v1v2 + u1v3v2 + u1v2v3 - u2v1v3) · Vector u
Expanding the dot product, we get:
= u2v3v1u1 - u3v2v1u1 - u3v1v2u1 + u1v3v2u1 + u1v2v3u1 - u2v1v3u1
Notice that all terms are the same as the terms we obtained on the left-hand side, just rearranged differently.
Therefore, we have shown that:
(Vector u · Vector v)^2 = [(Vector u × Vector v) × Vector v] · Vector u = u^2v^2