A line that meets the x-axis

Line l passes through A(1,2,−1)(1,2,-1) and B(2,a,1)(2,a,1). Given l cuts the x-axis, show a=−2a=-2. marks Find the shortest distance of O from l. marks

[modified 2008/NYJC/Promo/5]

Total marks: 8

Solution

Finding a

Intersection with an axis

Line ll passes through A(1,2,−1)A(1,2,-1) and B(2,a,1)B(2,a,1) and cuts the xx-axis. Show that a=−2a=-2.

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Hint

  1. Write l:r=OA→+λAB→l:\mathbf r=\overrightarrow{OA}+\lambda\overrightarrow{AB}.

  2. On the xx-axis y=0y=0 and z=0z=0.

Answer · Finding a

a=−2a=-2.

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The line meets the x-axis

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.

Shortest distance

Distance from a point to a line

Find the shortest distance of OO from ll.

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Hint

  1. Foot of the perpendicular: ON→⋅d=0\overrightarrow{ON}\cdot\mathbf d=0 with d=(1,−4,2)\mathbf d=(1,-4,2).

  2. Or use ∣OA→×d∣∣d∣\frac{|\overrightarrow{OA}\times\mathbf d|}{|\mathbf d|}.

Answer · Shortest distance

105/7\sqrt{105}/7; nearest point (10,2,−1)/7(10,2,-1)/7.

Explore the idea

Slide a point along l and measure its distance from O

O stays at the origin. Slide the trial point along l and follow its distance to O on the graph; the minimum occurs at the perpendicular foot.

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