11. Introduction to Three Dimensional Geometry

4 topics52 min

Chapter 11 of NCERT Mathematics for Class 11: Introduction to Three Dimensional Geometry. Three sections — Setting up space, Naming a point and Measuring between points. Four videos, 52 min in all.

Worked answers to the 13 exercise questions in this chapter: Exercise 11.1 · Exercise 11.2 · Miscellaneous Exercise

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Part 1

Setting up space

13 minThree perpendicular planes, and the eight regions they cut space intoThree planes through one point, each at right angles to the other two, cut space into eight regions - and eight is not automatic. Planes sharing a line cut it into six wedges instead.13 min · 13 parts

Part 2

Naming a point

13 minReading a triple as three perpendicular distances, one per planeThe first coordinate of a point in space is not its distance from the x-axis. It is the distance from the one plane that the x-axis does not lie in - the commonest confusion here.13 min · 13 parts12 minWhich coordinate goes to zero on which axis or planeA zero inside a triple of coordinates looks like nothing worth saying. It is the opposite - the most definite value a coordinate can carry, pinning the point onto a named plane.12 min · 12 parts

Part 3

Measuring between points

14 minApplying the right-triangle rule twice to get out of the planeThe distance formula in three dimensions looks like Pythagoras with an extra term stapled on. It is Pythagoras applied twice, in two planes meeting along one line.14 min · 14 parts

What you will be able to do

  • Explain why locating a fan tip or a hanging bulb inside a room takes three numbers rather than two, naming the room's floor and two adjacent walls as the three surfaces
  • State the two claims §11.3 has to prove, and say how one construction run in both directions supplies both
  • State what a single zero coordinate asserts about the point, in terms of a plane
  • Describe the box Fig 11.4 builds on the segment PQ and identify the two extra corners the derivation needs

What this chapter assumes you already know

  • Locating a point in a plane with two perpendicular axes, and reading a signed coordinate as a directed distance from an axis
  • The four quadrants of the plane and the sign pattern that names each one
  • That two distinct non-parallel planes meet in a line, and that a line can be perpendicular to a plane
  • Right angles between lines in space, and the idea of two lines being perpendicular without meeting
  • Counting by independent choices: two options taken three times over gives eight outcomes
  • Dropping a perpendicular from a point to a plane, and that the foot is unique
  • Dropping a perpendicular from a point to a line in a plane
  • That a line perpendicular to a plane is perpendicular to every line drawn in that plane through its foot

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.

  • "Three planes always make eight pieces."
  • "The axes come first and the planes are built from them."
  • "An octant is a corner of a box."
  • "Octants are numbered like binary digits."
  • "Roman numerals and letter names are two different classifications."
  • "The X-axis points to the right, as it does in a plane drawing."
  • "Space had to be described this way."
  • "x is the distance from the x-axis."