IIT JEE , AIEEE Forum - Mathematics , Vectors

avinash.sharma

Hello neeraj_agarwal_1990

I have used Bold+Underlined letters to represent vectors.

As we know the System of reciprocal vectors can be defined as:

p = (b x c)/[a b c] q = (c x a)/ [a b c] r = (a x b)/ [a b c]

But the given [ axp bxq cxr ] +[ axq bxr cxp ] + [ axr bxp cxq ] is always equal to 0 either p , q, r form reciprocal system or not. Let's see the proof ....

[ axp bxq cxr ] = (a x p) . { (b x q) x (c x r)}

= (a x p) . { [ b c r ] q - [q c r ] b }

= [a p q ] [ b c r ] - [a p b ] [q c r ] .......................(1)

Same way you will get

[ axq bxr cxp ] = (a x q) . { (b x r) x (c x p)}

= (a x q) . { [ b r p ] c - [b r c ] p }

= [a q c ] [ b r p ] - [a q p ] [b r c ] ..............................(2)

and

[ axr bxp cxq ] = {(a x r) x { (b x p)} . (c x q)}

= { [ a b p ] r - [r b p ] a } . (c x q)

= [a b p ] [ r c q ] - [r b p ] [a c q ] .....................(3)

Adding (1), (2) and (3) we get

[ axp bxq cxr ] +[ axq bxr cxp ] + [ axr bxp cxq ] = 0

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