For any vector r , what is ( r i )( r i )+( r j )( r j )+( r k )( r k ) equal to?

Step-by-step solution

( r i )( r i )+( r j )( r j )+( r k )( r k ) ...(i) We have, aligned r & =a i +b j +c k ( r i ) & =(a i +b j +c k ) i =a ( r j ) & =b ( r k ) & =c aligned and r i = | array lll i & j & k a & b & c 1 & 0 & 0 array |=- j (-c)-b k =c j -b k Similarly, r j =-c i +a k and r k =b i -a j Putting these values in (i) we get aligned & a( j -b k )+b(-c i +a k )+c(b i -a j ) &=a c j -a b k -b c i +a b k +b c i -a c j =0 aligned