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Question
If →α=a^i+b^j+c^k,→β=b^i+c^j+a^k and →γ=c^i+a^j+b^k be three coplanar vectors with →α≠→β≠→γ and →r=^i+^j+^k, then →r is perpendicular to
If →α=a^i+b^j+c^k,→β=b^i+c^j+a^k and →γ=c^i+a^j+b^k be three coplanar vectors with →α≠→β≠→γ and →r=^i+^j+^k, then →r is perpendicular to
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Solution
The correct option is C →γ
Given →α,→β,→γ are coplanar
⇒[→α→β→γ]=0
⇒∣∣
∣∣abcbcacab∣∣
∣∣=0
⇒(a+b+c)(a2+b2+c2−ab−bc−ca)=0
Hence, a+b+c=0
→r.→α=(→i+→j+→k).(a→i+b→j+c→k)=a+b+c=0 ⇒→r⊥→α
Similarly, →r⊥→β and →r⊥→γ
∴→r⊥(→α,→β,→γ)
Given →α,→β,→γ are coplanar
⇒[→α→β→γ]=0
⇒∣∣
∣∣abcbcacab∣∣
∣∣=0
⇒(a+b+c)(a2+b2+c2−ab−bc−ca)=0
Hence, a+b+c=0
→r.→α=(→i+→j+→k).(a→i+b→j+c→k)=a+b+c=0 ⇒→r⊥→α
Similarly, →r⊥→β and →r⊥→γ
∴→r⊥(→α,→β,→γ)
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