Three Dimensional GeometryIntroduction
The hill station near Riya's school has opened two new rides over the same valley. A zip line runs steeply down from a tower on one ridge, and a ropeway cable stretches across the valley a little lower. From the viewpoint the two cables seem to cross right in the middle, and the safety inspector is worried: if they really cross, a rider on the zip line could crash into the ropeway.
They don't cross. One cable passes over the other, like two pencils held one above the other and pointing different ways. But how close do they come? The inspector's rule is simple: two cables must never come within 2 m of each other.
You are the inspector. The blue cable is the zip line and the green one is the ropeway. Drag rider P along the zip line and rider Q along the ropeway, and watch the distance PQ. Drag the empty sky sideways to turn the valley and see it from another side.
The cables never come closer than 3 m, so the rides pass the inspection. Look at the two little squares: at the closest pair, PQ is perpendicular to both cables. Finding that pair took you some patient dragging. By the end of this chapter you will get the same 3 m in a few lines of working, from one formula.
What Fixes a Line?
In Class XI you located points in space with three
Suppose the inspector has to describe the zip line to an engineer over the phone. Which of these is enough to pin the line down completely?
A point and a direction fix a line. Two points do the job too, since two points give you a direction. So here is the plan for the chapter. First we learn to write the direction of a line as numbers: direction cosines and direction ratios. Then we write the equation of a line, in vector form and in cartesian form. Then we find the angle between two lines. Finally we find the shortest distance between two lines, including lines like our two cables, which neither meet nor run side by side.