Skew lines and a perpendicular foot

Lines l1,l2l_1,l_2 are r=(1,0,1)+λ(1,1,−2)\mathbf r=(1,0,1)+\lambda(1,1,-2) and r=(1,2,−1)+μ(7,5,1)\mathbf r=(1,2,-1)+\mu(7,5,1) respectively.

(i) Show they are skew lines. marks

(ii) Find the position vector of the foot of perpendicular from A(1,2,−1)(1,2,-1) to l1l_1. marks

[modified 2009/NYJC/Prelim/I/3]

Total marks: 6

Solution

(i)

Skew lines

l1:r=(1,0,1)+λ(1,1,−2)l_1:\mathbf r=(1,0,1)+\lambda(1,1,-2) and l2:r=(1,2,−1)+μ(7,5,1)l_2:\mathbf r=(1,2,-1)+\mu(7,5,1). Show they are skew.

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Hint

  1. Check the directions are not parallel.

  2. Equate two coordinates, solve, and test the third.

Answer · (i)

Nonparallel directions and no common point: the first two coordinates require μ=1,λ=7\mu=1,\lambda=7, but the third coordinates disagree.

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Two lines that never meet

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

(ii)

Perpendicular foot

Find the foot of the perpendicular from A(1,2,−1)A(1,2,-1) to l1l_1.

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Hint

  1. NN on l1l_1: ON→=(1,0,1)+λ(1,1,−2)\overrightarrow{ON}=(1,0,1)+\lambda(1,1,-2).

  2. AN→⋅(1,1,−2)=0\overrightarrow{AN}\cdot(1,1,-2)=0.

Answer · (ii)

Foot (2,1,−1)(2,1,-1).

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Move A and follow its perpendicular foot on l₁

Drag a coordinate slice to move A. N always stays on l₁, with AN perpendicular to its direction (1, 1, −2). Reset restores A = (1, 2, −1) and N = (2, 1, −1).

Open the interactive view to explore this geometry in 3D.