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Question:

Let L1 and L2 denote the lines →r = i^ + λ(-i^ + 2j^ + 2k^), λ ∈ R and →r = μ(2i^ - j^ + 2k^), μ ∈ R respectively. If L3 is a line which is perpendicular to both L1 and L2 and cuts both of them, then which of the following options describe(s) L3?

r→=13(2i^+k^)+t(2i^+2j^−k^),tϵR

r→=29(4i^+j^+k^)+t(2i^+2j^−k^),tϵR

r→=29(2i^−j^+2k^)+t(2i^+2j^−k^),tϵR

r→=t(2i^+2j^−k^),tϵR

Solution:

Correct option is C. →r = 29(2i^ - j^ + 2k^) + t(2i^ + 2j^ - k^), t ∈ R
Both given lines are skew lines. So direction ratios of any line perpendicular to these lines are 6i^ + 6j^ - 3k^ <2, 2, -1;>
Points at shortest distance between given lines are A, B.
→AB ⊥ line L1
→AB ⊥ line L2
So A(8/9, 2/9, 2/9)
Now equation of required line →r = (8/9i^ + 2/9j^ + 2/9k^) + α(2i^ + 2j^ - k^)
Now by option B, C, D are correct.