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Question

Let a=2i+^j+^k, b=^i+2^j^k and a unit vector c be coplanar. If c is perpendicular to a, then c is equal to

A
12(^j+^k)
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B
13(^i^j^k)
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C
15(^i2^j)
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D
15(^i^j^k)
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Solution

The correct option is A 12(^j+^k)
It is given that c is coplanar with a and b, we take c=pa+qb (i)
where, p and q are scalars.
Since, cac.a=0
Taking dot product of a in Eq. (i), we get
c.a=pa.a+qb.a0=p|a|2+q|b.a|⎢ ⎢ ⎢ ⎢ ⎢a=2^i+^j+^k|a|=22+1+1=6.a.b=(2^i+^j+^k).(^i+2^j^k)=2+21=3⎥ ⎥ ⎥ ⎥ ⎥
0=p.6+q.3q=2p
On putting in Eq. (i), we get
c=pa+b(2p)c=pa2pbc=p(a2b)c=p[(2^i+^j+^k)2(^i+2^j^k)]c=p(3^j+3^k)|c|=p(3)2+32|c|2=p2(18)2|c|2=p2.181=p2.181=p2.18 [|c|=1]p2=118p=±132c=±1(^j+^k2

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