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Question: If the given vectors \(( - bc,b^{2} + bc,c^{2} + bc),\) \((a^{2} + ac, - ac,c^{2} + ac)\) and \((a^...

If the given vectors (−bc,b2+bc,c2+bc),( - bc,b^{2} + bc,c^{2} + bc),

(a2+ac,−ac,c2+ac)(a^{2} + ac, - ac,c^{2} + ac) and (a2+ab,b2+ab,−ab)(a^{2} + ab,b^{2} + ab, - ab) are coplanar, where none of a, b and c is zero, then

A

a2+b2+c2=1a^{2} + b^{2} + c^{2} = 1

B

bc+ca+ab=0bc + ca + ab = 0

C

a+b+c=0a + b + c = 0

D

a2+b2+c2=bc+ca+aba^{2} + b^{2} + c^{2} = bc + ca + ab

Answer

bc+ca+ab=0bc + ca + ab = 0

Explanation

Solution

Accordingly, $\left| \begin{matrix}

  • bc & b^{2} + bc & c^{2} + bc \ a^{2} + ac & - ac & c^{2} + ac \ a^{2} + ab & b^{2} + ab & - ab \end{matrix} \right| = 0$

⇒(ab+bc+ca)3=0⇒ab+bc+ca=0.\Rightarrow (ab + bc + ca)^{3} = 0 \Rightarrow ab + bc + ca = 0.