Let →a,→b and →c be mutually perpendicular vectors of the same magnitude.If a vector →x satisfies the equation →a×[(→x−→b)×→a]+→b×[(→x−→c)×→b]+→c×[(→x−→a)×→c]=0, then →x is given by
A
12(→a+→b+→c)
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B
13(2→a+→b+→c)
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C
13(→a+→b+→c)
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D
12(→a+→b−2→c)
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Solution
The correct option is A12(→a+→b+→c) Let ∣∣→a∣∣=∣∣∣→b∣∣∣=∣∣→c∣∣=λ, then a2(→x−→b)−[→a.(→x−→b)]+b2(→x−→c)−[→b.(→x−→c)]→b+c2(→x−→a)−→c(→x−→a]→c=0 λ2(3→x−→a−→b−→c)−[(→a.→x)→a−(→b.→x)→b−(→c.→x)→c]−(→a.→b)→a+(→b.→c)→b+(→c.→a)→c=0 ⇒λ2(3→x−→a−→b−→c)−λ2→x=0 ⎡⎢
⎢
⎢
⎢⎣∴→a=→a.→x∣∣→a∣∣2→a+→b.→x∣∣∣→b∣∣∣2→b+→c.→x∣∣→c∣∣2→c⎤⎥
⎥
⎥
⎥⎦ ⇒→x=12(→a+→b+→c)