Question

If a + b + c = 0 then prove that a × b = b × c = c × d.

Answer 1

Given that

a + b + c = 0

Case I : -

a + b + c = 0

Taking Cross Product with 'a' on both sides, we get

a × (a + b + c) = 0

a×a + a×b + a×c = 0

a × b  +  a × c = 0           ∵ a×a = 0

a × b = -  a × c

a × b = c × a                  ∵    -  a × c = c × a

a × b = c × a     ....... (i)

Case II : -

a + b + c = 0

Taking Cross Product with 'b' on both sides, we get

b × (a + b + c) = 0

b×a + b×b + b×c = 0

b × a   +   b × c = 0           ∵ b×b = 0

b × c = -  b × a

b × c = a × b                  ∵     -  b × a = a × b

a × b = b × c     ....... (ii)

From (i) & (ii), we have

a × b = c × a      &    a × b = b × c

 ⇒  a × b = b × c = c × a             As Required