11. Three Dimensional Geometry

5 topics2 h

Chapter 11 of NCERT Mathematics for Class 12: Three Dimensional Geometry. Three sections — Pinning down a direction in space, The equation of a line and Two lines at once. Five videos, 2 h in all.

Worked answers to the 25 exercise questions in this chapter: Exercise 11.1 · Exercise 11.2 · Miscellaneous Exercise

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Part 1

Pinning down a direction in space

19 minThree angles with the axes, and why their cosines square up to one19 min · 15 parts16 minDirection ratios as any proportional triple, and normalising back to cosines16 min · 13 parts

Part 2

The equation of a line

20 minOne point plus one direction fixes a line, in vector form and in the symmetric Cartesian form the parameter eliminates to20 min · 16 parts

Part 3

Two lines at once

17 minThe angle between two lines, from cosines or from ratios17 min · 15 parts29 minSkew lines, what shortest distance can mean when two lines never meet, and computing it in both the skew and parallel cases29 min · 20 parts

What you will be able to do

  • Draw a directed line through the origin and mark the three angles it makes with the positive axes
  • Define a direction-ratio triple as any triple proportional to the direction cosines, and state the scaling relation the chapter writes
  • State the two conditions the chapter says are each enough to fix a line, and say which of the two the chapter goes on to develop
  • Explain why two lines that never meet still have a well-defined angle, and what construction supplies it
  • Sort two lines in space into the three cases the chapter names, and say which distance each case yields

What this chapter assumes you already know

  • Rectangular axes in space, the coordinates of a point, and the eight regions the three coordinate planes cut space into, from Class XI
  • The distance between two points in space, from Class XI
  • Cosine of an angle between zero and one hundred eighty degrees, and its sign on each side of a right angle
  • The cosine of a supplement, from Class XI trigonometry
  • A right-angled triangle, and the cosine as adjacent over hypotenuse
  • Chapter 10 of this book, which introduces the direction angles and their cosines for a vector before this chapter recalls them
  • Direction angles and direction cosines, and the identity that their squares total one — the previous topic
  • Proportional triples, and what it means for two triples to be in the same ratio

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The three direction angles are independent, so I can pick any three."
  • "Equal angles with the three axes means each cosine is one third."
  • "Reversing the line flips one of the signs."
  • "A direction cosine can be bigger than one."
  • "A line not through the origin has no direction cosines."
  • "The angle in the triangle is gamma because it looks like it."
  • "A negative direction cosine is a mistake."
  • "Direction cosines belong to a segment, so they change if I extend it."