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CBSE · Class 11 · Mathematics

Introduction to Three Dimensional Geometry (Ch 11)

3 concepts · 24 practice questions with answers · Free to read

Axes and planes in space

From daily life

A room corner has three edges meeting at right angles: floor edge along the wall, the other floor edge, and the vertical edge. These are the x, y and z axes.

Simple definition

In space we use three mutually perpendicular lines through the origin O: the x-axis, y-axis and z-axis. Each pair of axes gives a coordinate plane: XY, YZ and ZX. The three planes divide space into 8 octants.

Rule 1: Axes and planes — Name the plane by the two axes it contains.

  • Easy: The plane containing the x- and y-axes is the XY-plane (z = 0).
  • Medium: The plane containing the y- and z-axes is the YZ-plane (x = 0).
  • Exam-type: The plane containing the z- and x-axes is the ZX-plane (y = 0).

Rule 2: Octants — Signs of x, y, z decide the octant.

  • Easy: (+, +, +) is the first octant.
  • Medium: There are 2 × 2 × 2 = 8 octants.
  • Exam-type: A point on a plane or axis does not belong to any octant.
Remember

Right-handed system: turn from x towards y and z points up in the direction a screw would move.

Common mistake

Thinking space has 4 parts like the plane. It has 8.

Memory clue

Three axes, three planes, eight octants.

Practice questions

Easy
  1. 1How many coordinate planes are there?
  2. 2What is the equation of the XY-plane?
  3. 3What is the equation of the YZ-plane?
Medium
  1. 4How many octants are there?
  2. 5In which octant is (2, 3, 4)?
  3. 6What is the equation of the ZX-plane?
Challenging
  1. 7Which axis do you travel along from (0, 0, 0) to (0, 0, 5)?
  2. 8In which octant is a point with x < 0, y > 0, z > 0?
Show answers (try them yourself first)
  1. 13
  2. 2z = 0
  3. 3x = 0
  4. 48
  5. 5First octant (all positive)
  6. 6y = 0
  7. 7The z-axis
  8. 8The octant (−, +, +)

Coordinates of a point in space

From daily life

To locate a bird you give three numbers: how far east, how far north and how high.

Simple definition

A point P is described by (x, y, z): x is the signed distance from the YZ-plane, y from the ZX-plane and z from the XY-plane. Points on the x-axis are (x, 0, 0); on the XY-plane (x, y, 0).

Rule 1: Where is the point — Look at which coordinates are zero.

  • Easy: (3, 0, 0) is on the x-axis.
  • Medium: (0, 4, 5) is on the YZ-plane.
  • Exam-type: (2, 3, 0) is on the XY-plane.

Rule 2: Distance from planes — Absolute value of the matching coordinate.

  • Easy: P(3, 4, 5): distance from the XY-plane is 5.
  • Medium: P(3, 4, 5): distance from the YZ-plane is 3.
  • Exam-type: P(−2, 6, −1): distance from the ZX-plane is 6.
Remember

The sign tells the side of the plane; the distance is the absolute value.

Common mistake

Mixing up which coordinate is the height.

Memory clue

z is the zenith (height).

Practice questions

Easy
  1. 1Where does (0, 0, 7) lie?
  2. 2Where does (4, 5, 0) lie?
  3. 3What is the distance of (2, −3, 6) from the XY-plane?
Medium
  1. 4What is the distance of (2, −3, 6) from the ZX-plane?
  2. 5What is the foot of the perpendicular from (3, 4, 5) to the XY-plane?
  3. 6What is the foot of the perpendicular from (3, 4, 5) to the x-axis?
Challenging
  1. 7Write the image of (2, 3, 4) in the XY-plane.
  2. 8A box has one corner at the origin and the opposite corner at (4, 3, 2). Write the other corners.
Show answers (try them yourself first)
  1. 1On the z-axis
  2. 2On the XY-plane
  3. 36
  4. 43
  5. 5(3, 4, 0)
  6. 6(3, 0, 0)
  7. 7(2, 3, −4)
  8. 8(4,0,0), (0,3,0), (0,0,2), (4,3,0), (4,0,2), (0,3,2)

Distance between two points

From daily life

A fly goes from one corner of a room to the opposite corner. Its straight distance comes from Pythagoras twice.

Simple definition

The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. The distance of P(x, y, z) from the origin is √(x² + y² + z²).

Formula: PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

Rule 1: Distance between points — Subtract coordinates, square, add, root.

  • Easy: P(1, 2, 3), Q(4, 6, 15): 13.
  • Medium: P(0, 0, 0), Q(2, 3, 6): 7.
  • Exam-type: P(1, 1, 1), Q(3, 3, 2): 3.

Rule 2: Use the formula — Check a triangle or an unknown coordinate.

  • Easy: A(0, 0, 0), B(3, 0, 0), C(0, 4, 0): AB = 3, AC = 4, BC = 5, a right triangle.
  • Medium: Points on the x-axis (a, 0, 0) at distance 5 from (1, 0, 0): a = 6 or −4.
  • Exam-type: Collinear: A(1, 2, 3), B(2, 4, 6), C(3, 6, 9): AB + BC = AC.
Remember

Distance formula in a plane is the special case z₁ = z₂.

Common mistake

Forgetting to square the z difference.

Memory clue

3D Pythagoras: add a z² under the root.

Practice questions

Easy
  1. 1Find the distance of (3, 4, 12) from the origin.
  2. 2Find the distance between (1, 2, 3) and (4, 6, 15).
  3. 3Find the distance between (0, 0, 0) and (1, 1, 1).
Medium
  1. 4Find the distance between (2, −1, 3) and (−2, 2, 3).
  2. 5Show that (1, 2, 3), (2, 3, 4), (3, 4, 5) are collinear.
  3. 6Find y if the point (0, y, 0) is at distance √26 from (3, 0, 4).
Challenging
  1. 7The diagonal of a cube of side 4. Find its length.
  2. 8Show that AB = AC for A(0, 0, 0), B(2, 2, 1), C(−1, 2, −2).
Show answers (try them yourself first)
  1. 113
  2. 213
  3. 3√3
  4. 45
  5. 5AB = BC = √3 and AC = 2√3 = AB + BC
  6. 69 + y² + 16 = 26, so y = 1 or −1
  7. 7√(16 + 16 + 16) = 4√3
  8. 8AB = 3, AC = 3: AB = AC = 3