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Let l1a n dl2 be the two skew lines. If ...

Let `l_1a n dl_2` be the two skew lines. If `P ,Q` are two distinct points on `l_1n dR , S` are two distinct points on `l_2,` then prove that `P R` cannot be parallel to `Q Sdot`

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Let equation of the line `l_(1)` be `vecr=veca+lamdavecb` and equation of the line `l_(2)` be `vecr=vecc+muvecd`,
where `veca-vecc, vecb and vecd` are non-coplanar.
Let the position vectors of points P and Q be `veca+lamda_(1)vecb and vecaa+lamda_(2)vecb`, respectively.
Let the position vectors of points R and S be `vecc+mu_(1)vecd and vecc+mu_(1)vecd`, respectively.
Then the lines PR and QS are parallel if and only if `vecc-veca+mu_(1)vecd-lamda_(1)vecb=k(vecc-veca+mu_(2)vecd-lamda_(2)vecb)`
i.e., `" "(1-k)(vecc-veca)+(mu_(1)-kmu_(2))vecd - (lamda_(1)-klamda_(2))vecb=veco`
`therefore" "1-k=0, mu_(1)-kmu_(2)=0, lamda_(1)-klamda_(2)=0`
i.e., `" "mu_(1)=mu_(2) and lamda_(1)=lamda_(2)` which is not possible
Therefore, PR can not be parallel to QS.
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