Three lines L_1:r→=λi^, λ∈R, L_2:r→=k^+μj^, μ∈R and L_3: r→=i^+j^+νk^, ν∈R are given. For which point(s) Q and L_2 can we find a point P on L_1 and a…

Question

Three lines L_1:r→=λi^, λ∈R, L_2:r→=k^+μj^, μ∈R and L_3: r→=i^+j^+νk^, ν∈R are given. For which point(s) Q and L_2 can we find a point P on L_1 and a point R on L_3 so that P, Q and R are collinear?

MARKS Content Team · Accepted Answer

Correct answer: (C) k^+1/2j^; (D) k^-1/2j^. As given L_1⇒r→=λi, λ∈R L_2⇒r→=k+μj, μ∈R and L_3⇒r→=i+j+νk, ν∈R Let point Pλ, 0, 0, Q0, μ, 1 and R1, 1, ν and PQ→=-λi^+μj^+k^, PR→=1-λi^+j^+νk^ as P, Q and R are co-linear ⇒PQ→ is parallel to PR→ ⇒-(λ)/(1-λ)=(μ)/1=1/(ν) ⇒λ=(μ)/(μ-1) and ν=1/(μ) hence μ≠0 μ≠1 Q≠0, 0, 1 and Q≠0, 1, 1 Q≠k^ & Q≠j^+k^

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Three linesL1:r→=λi^, λ∈R,L2:r→=k^+μj^, μ∈R andL3: r→=i^+j^+νk^, ν∈Rare given. For which point(s) Q and L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear?

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