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Question

If the vectors (bc,b2+bc,c2+bc),(a2+ac,ac,c2+ac) and (a2+ab,b2+ab,ab) are coplanar (where none of a, b or c is zero). Then?

A
a2+b2+c2=1
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B
a+b+c=0
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C
ab+bc+ca=0
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D
a2+b2+c2=(a+b+c)2
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Solution

The correct option is C ab+bc+ca=0

solve:- we have to Given three
vectors which are coplaner
so, their scaler triple product
must be zero.
∣ ∣ ∣bcb2+bcc2+bca2+acacc2+aca2+abb+abab∣ ∣ ∣=0

R1aR1R2bR2,R3CR3

1abc∣ ∣ ∣abca(b2+bc)a(c2+bc)b(a2+ac)abcb(c2+ac)c(a2+ab)c(b+ab)abc∣ ∣ ∣=0

∣ ∣ ∣bca(b+c)a(c+b)b(a+c)acb(c+a)c(a+b)c(b+a)ab∣ ∣ ∣=0

R1R1+R2+R3

∣ ∣ ∣ab+bc+caab+bc+caab+bc+cab(a+c)acb(c+a)c(a+b)c(b+a)ab∣ ∣ ∣=0

(ab+bc+ca)∣ ∣ ∣1.11b(a+c)acb(c+a)c(a+b)c(a+b)ab∣ ∣ ∣=0

c1c1c3,c2c2c3

(ab+bc+ca)∣ ∣0.010(ac+bc+ba)b(c+a)ac+bc+abbc+ac+abab∣ ∣=0

(ab+bc+ca)(0+(ab+bc+ca)2)=0

(ab+bc+ca)3=0

ab+bc+ca=0

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