Let the line of the shortest distance between the lines L1:r=(i^+2j^+3k^)+λ(i^−j^+k^) and L2:r=(4i^+5j^+6k^)+μ(i^+j^−k^) intersect L1 and L2 at P and Q respectively. If (α,β,γ) is the midpoint of segment PQ, then 2(α+β+γ) is equal to:
Answer: 21
Solutions · 1
Official solution
P=(1+λ,2−λ,3+λ), Q=(4+μ,5+μ,6−μ); PQ=(3+μ−λ,3+μ+λ,3−μ−λ).
Perpendicular to (1,−1,1): 3λ+μ=3. Perpendicular to (1,1,−1): λ+3μ=−3.
Solving: λ=23,μ=−23.
\[P=(2.5,0.5,4.5),\qquad Q=(2.5,3.5,7.5).\]
\[(\alpha,\beta,\gamma)=(2.5,2,6).\]
\[2(\alpha+\beta+\gamma)=2(10.5)=21.\]