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Question

# Let →p,→q,→r be three mutually perpendicular vectors of the same magnitude If a vector →x satisfies the equation →p×((→x−→q)×→p)+→q×((→x−→r)×→q)+→r×((→x−→p)×→r)=0 is given by

A
12(p+q2r)
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B
13(p+q+r)
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C
12(p+q+r)
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D
13(2p+qr)
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Solution

## The correct option is C 12(→p+→q+→r)→p×((→x−→q)×→p)+→q×((→x−→r)×→q)+→r×((→x−→p)×→r)=0⇒(→p.→p)(→x−→q)−(→p.(→x−→q)→p)+(→q.→q)(→x−→r)−(→q.(→x−→r)→q)+→r.→r(→x−→p)−(→r(→x−→p)→r)=0⇒λ2Σ(→x−→q)−Σ(→p.→x→p)=0Where→p.→p=→q→q=→r→r=λ2(Since→p.→q=→q→r=→r→p=0)⇒λ2[3→x−(→p+→q+→r)]−Σ{(→p→x)→p}=0[∵Σ(→p→x)→p=→xλ2]⇒2→x=→p+→q+→r⇒→x=→p+→q+→r2

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