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Question

If l1,m1,n1 and l2,m2,n2 are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be

A
(m1n2m2n1),(n1l2l1n2),(l1m2l2m1)
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B
(l1l2m2m1),(m1m2n1n2),(n1n2l2l1)
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C
1l21+m21+n21,1l22+m22+n22,13
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D
13,13,13
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Solution

The correct option is A (m1n2m2n1),(n1l2l1n2),(l1m2l2m1)
Let lines are l1x+m1y+n1z+d=0 (i)
and l2x+m2y+n2z+d=0 (ii)
If lx+my+nz+d=0 is perpendicular to (i) and (ii), then,
ll1+mm1+nn1=0,ll2+mm2+nn2=0
lm1n2m2n1=mn1l2l1n2=nl1m2l2m1=d
Therefore, direction cosines are
(m1n2m2n1),(n1l2l1n2),(l1m2l2m1).

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