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Question

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

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

## The correct option is B 12(→p+→q+→r)To make the problem easier, let ¯¯¯p=^i,¯¯¯q=^j,¯¯¯r=^k.LHS becomes (¯¯¯x−^j)−^i.(¯¯¯x−^j)^i+(¯¯¯x−^k)−^j.(¯¯¯x−^k)^j+(¯¯¯x−^i)−^k.(¯¯¯x−^i)^k∴0=3(a^i+b^j+c^k)−^i−^j−^k−a^i−b^j−c^k∴^i+^j+^k=2a^i+2b^j+2c^kComparing co-efficients, we get a=b=c=12Thus, ¯¯¯x=12(¯¯¯p+¯¯¯q+¯¯¯r)

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