Please only answer if you are 100% sure Please answer with details Show that the

Question

Please only answer if you are 100% sure Please answer with details
Show that the following relation holds for any three vectors A, B, and C: Ax(BxC)=(A.C)B-C(A.B) "x" and "." Stand for cross and dot products, respectively. Please only answer if you are 100% sure Please answer with details
Show that the following relation holds for any three vectors A, B, and C: Ax(BxC)=(A.C)B-C(A.B) "x" and "." Stand for cross and dot products, respectively. Please only answer if you are 100% sure Please answer with details
Ax(BxC)=(A.C)B-C(A.B) "x" and "." Stand for cross and dot products, respectively.

Explanation / Answer

B×C is a vector perpendicular to the plane formed by B and C. Hence the vector A×(B×C) lies in the plane formed by B and C.
So we can write A×(B×C) as,

A×(B×C) = BC†

Multiply(dot product) † by A to get,

A.(A×(B×C)) = (A.B) (A.C)

Since

A.(A×(B×C)) = (A×A).(B×C)=0

We get (A.B) (A.C)=0

A.C = A.B =

So

A×(B×C) = ((A.C)B (A.B)C)

The above equation is an identity and holds good for any three vectors A,B and C.

So, taking the three vectors as i,i and j as A, B and  C respectively, we get

j = (j)

= 1

So
A×(B×C) = (A.C)B (A.B)C

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