Introduction to Three Dimensional Geometry
Class 11 · Mathematics
4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.
Coordinates and Octants
Part of: Coordinates and Distance
Three perpendicular planes frame space; coordinates measure wall distances; sign patterns sort eight octants; coordinate planes zero one coordinate.
Why it matters. Triple numbers pin space points. Octants organise signs.
Formulas
- The three coordinate planes XY, YZ and ZX divide space into eight octants.
- Any point on the x-axis is (x, 0, 0); on the y-axis (0, y, 0); on the z-axis (0, 0, z).
Key words
- Coordinate axis
- : The three mutually perpendicular lines used as reference axes.
- Coordinate plane
- : A flat surface fixed by two axes, such as the XY plane where z is zero.
- Octant
- : One of the eight regions into which the planes divide space.
- Origin
- : The point (0, 0, 0) where all three axes meet.
Try it in two minutes
- Read x, y, z walls.
- Check sign pattern.
- Name the octant.
Try at home. Locate your room corner.
Distance Between Points
Part of: Coordinates and Distance
Square each coordinate gap and root the sum; origin distance roots squares directly; equal gaps test special triangles.
Why it matters. 3D distance sizes space. Gaps verify shapes.
Formulas
- PQ = root of (x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2.
- A point (x, y, z) is root of x^2 + y^2 + z^2 away from the origin.
Key words
- Distance formula
- : The root of squared coordinate gaps that measures separation in space.
- Origin distance
- : The root of x^2 + y^2 + z^2 measuring separation from (0, 0, 0).
- Collinear points
- : Points lying on one straight line, tested by adding smaller gaps.
Try it in two minutes
- Subtract coordinates.
- Square and add.
- Root the total.
Try at home. Measure room diagonal.
Section Formula in 3D
Part of: Section Formula and Direction
Internal division weights m-x2-plus-n-x1 over m-plus-n per coordinate; external flips one sign; midpoints halve sums; centroid averages vertices.
Why it matters. Division locates interior points. Midpoints halve work.
Formulas
- Internal division: ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n)).
- External division flips one sign: use (m x2 - n x1)/(m - n) per coordinate.
- Midpoint halves the sums: ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2).
- Triangle centroid averages vertices: ((x1 + x2 + x3)/3, ...).
Key words
- Section formula
- : Splitting a segment in ratio m:n; external division flips one sign.
- Midpoint
- : The point halving a segment, found by averaging endpoints.
- Centroid
- : The average of three vertices marking a triangle centre.
- Internal and external
- : Division inside the segment versus on its extension outside.
Try it in two minutes
- Weight by opposite ratio.
- Flip sign outside.
- Average for midpoints.
Try at home. Find midpoint of room diagonal.
Direction Cosines and Ratios
Part of: Section Formula and Direction
Direction cosines project unit length onto axes and square-sum to one; ratios scale freely; gaps over distance give cosines directly.
Why it matters. Cosines aim lines in space. Ratios skip normalising.
Formulas
- Direction cosines satisfy l^2 + m^2 + n^2 = 1.
- l = (x2 - x1)/PQ, m = (y2 - y1)/PQ, n = (z2 - z1)/PQ for a line joining two points.
- Direction ratios scale freely; divide by root of a^2 + b^2 + c^2 to get cosines.
Key words
- Direction cosine
- : The cosines of the angles a line makes with the x, y and z axes.
- Direction ratio
- : Any triple proportional to the direction cosines of a line.
- Unit projection
- : The unit-length projection of a line direction on each axis.
Try it in two minutes
- Find coordinate gaps.
- Divide by distance.
- Square-sum to check.
Try at home. Aim a torch, read angles.
What your child will practise
- Coordinates and Octants: Jungle Puzzle (Clue Puzzle)
- Coordinates and Octants: Jungle Quick Fire (Quick Fire)
- Direction Cosines and Ratios: Candy Puzzle (Clue Puzzle)
- Distance Between Points: Desert Quest (Flip and Match)
- Section Formula in 3D: Space Quick Fire (Quick Fire)
- Quiz: Coordinates and Octants (8 questions)
- Quiz: Direction Cosines and Ratios (8 questions)
- Quiz: Distance Between Points (8 questions)
- Quiz: Section Formula in 3D (8 questions)