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CBSE Class 11 · Mathematics

Introduction to Three Dimensional Geometry

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Introduction to Three Dimensional Geometry” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 11
  • Medium level
  • 8 concepts

1Coordinate Axes and Coordinate Planes in Three Dimensional Space

Three-dimensional space needs three axes through an origin O: x, y, z, pairwise perpendicular as taught. Three coordinate planes (xy, yz, zx) split space into octants. A plane here is a wall of the room-story, not a 2-d chapter by itself.

A page with only x and y is not this space.

Figure. Three signed axes meet at O. The coordinate planes are XY (z = 0), YZ (x = 0) and ZX (y = 0). This is a first-octant schematic, not a measured isometric.

How it works

  1. Draw three axes through OThe frame.
  2. Name the three planesxy, yz, zx.
  3. Keep a point needing three numbers3-d.

2Coordinates of a Point in Space

A point in space is an ordered triple (x, y, z). (2, −1, 4) is not (2, 4, −1). The third number is not optional decoration. A point on the xy-plane has z=0.

Dropping z because the paper is flat is a miss.

Figure. A point in space is the ordered triple (x, y, z). Each coordinate is the signed distance from P to one coordinate plane. Dashed drops are the schematic feet, not a measured box.

How it works

  1. Write (x, y, z) in orderThe triple.
  2. Read which plane if a coordinate is 0The sit.
  3. Refuse a pair as a space-pointThree numbers.

3Distance between Two Points

Distance between (x1,y1,z1) and (x2,y2,z2) is √[(Δx)²+(Δy)²+(Δz)²]. From (1,2,2) to (3,3,6) is √(4+1+16)=√21. A 2-d distance that ignores z is a different (wrong) length.

3+1+4=8 is the walk-steal.

Figure. Distance in space is the root of the three squared steps: d = √(Δx² + Δy² + Δz²). The three dashed legs are a plane schematic of those steps, not a measured 3-D box.

How it works

  1. Subtract each coordinateThe three legs.
  2. Square, add, square-rootThe space-distance.
  3. Keep all three squaresIncluding Δz.

Space 3-4-ish

Distance from (1, 2, 2) to (3, 3, 6).

  • Δx, Δy, Δz2, 1, 4
  • Squares4+1+16=21
  • Distance√21

Pro tip. Three legs; do not drop z.

4A definition is a test you can run

Triple / space-distance is a test: three numbers plus the three-square root. If you only say “3-d”, you have a heading.

A room-photo is not the test.

Figure. The definition is a test: x, y and z are the signed distances from P to the planes x = 0, y = 0 and z = 0. A picture without those three measures is not yet a space point.

How it works

  1. Name the two triplesThe object.
  2. Run √(Δx²+Δy²+Δz²)The test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given are the triples. The unknown is a plane-sit or the distance. Copy z before you subtract only x and y.

Using (2,−1) as if z were missing when z=4 was given is a silent given.

Figure. Write the two triples first. Distance is the unknown those six numbers determine. Naming d before the coordinates is backwards.

How it works

  1. Copy both triplesThe given.
  2. Name the sit or the lengthThe unknown.
  3. Then subtract three waysGiven first.

6One worked case is enough at this class

One distance is enough. A page of the same √21 clone does not add a new octant.

Ten twins of (2,−1,4) are still one idea.

Figure. Worked case: O to P(1, 2, 2). 1² + 2² + 2² = 9, so d = 3. Labels carry the three steps; the path is a plane schematic.

How it works

  1. Compute one space-distanceThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A room-photo is a setting. The maths is (x,y,z). A corner is not O until you agree axes.

A box still is not √(Δx²+Δy²+Δz²).

Figure. Calling a flat (x, y) sketch “3-D geometry” swaps the school name for the picture. Space needs a third axis. Both panels are 2-D drawings; only the right one names three coordinates.

How it works

  1. Keep the room as a storyA setting.
  2. Write the triplesThe maths.
  3. Then the distanceDo not swap.

8Check by the opposite action or the opposite test

Check a distance by swapping the two points — it must match — or by a 3-d Pythagoras rebuild. Dropping Δz must change the answer if Δz≠0.

A 2-d check that ignores z is not the opposite test.

Figure. Opposite test for a midpoint: AM and MB must match, and AM + MB must equal AB. If the halves disagree, M is not the midpoint of AB.

How it works

  1. State √21The claim.
  2. Swap ends or re-square all threeThe check.
  3. Mend if they disagreeHonest.
Distance from (1,2,2) to (3,3,6) is
  1. √21
  2. √5
  3. 7

4+1+16=21; dropping z would be √5.

Notes

  • The official chapter title is “Introduction to Three Dimensional Geometry”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • distance = √[(x2−x1)²+(y2−y1)²+(z2−z1)²]

Recap

Hold these pegs from the official chapter “Introduction to Three Dimensional Geometry”. The wording is ExamMaster’s teaching, not a textbook recap.

Coordinate Axes and Coordinate Planes in Three Dimensional Space
Three-dimensional space needs three axes through an origin O: x, y, z, pairwise perpendicular as taught.
Coordinates of a Point in Space
A point in space is an ordered triple (x, y, z).
Distance between Two Points
Distance between (x1,y1,z1) and (x2,y2,z2) is √[(Δx)²+(Δy)²+(Δz)²].
A definition is a test you can run
Triple / space-distance is a test: three numbers plus the three-square root.
Name the given before the unknown
The given are the triples. The unknown is a plane-sit or the distance. Copy z before you subtract only x and y.
One worked case is enough at this class
One distance is enough. A page of the same √21 clone does not add a new octant.

Practise Introduction to Three Dimensional Geometry

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