Schooling · Class 11 · Mathematics · Introduction to Three Dimensional Geometry · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Introduction to Three Dimensional Geometry: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Three mutually perpendicular axes…
- Any point P in space is represented…
- The coordinate planes are the…
- Distance in three dimensions follows…
- The section formula finds a point…
- Signs of coordinates decide which…
- Formulas and facts
- Distance between two points
- Distance from the origin
- Section formula (internal division)
- Section formula (external division)
- Midpoint
- Centroid of a triangle
- Watch out for
- Forgetting to square all three…
- Using the wrong sign in external…
- Placing a point in the wrong octant…
- Taking the centroid formula with…
- Writing (x, y) instead of (x, y, z)…
What this chapter is about
This chapter extends the coordinate geometry you learnt in two dimensions to three-dimensional space. In Class 10 and earlier Class 11 work, you located points on a plane using two numbers (x, y). Real objects — a bird in the sky, a corner of a room, a satellite above Earth — need three numbers because they occupy space, not just a flat surface. Here you learn how to describe such positions precisely.
Key ideas
- Three mutually perpendicular axes — the x-axis, y-axis and z-axis — meet at the origin O, dividing space into eight octants.
- Any point P in space is represented by an ordered triple (x, y, z), giving its signed distances from the three coordinate planes.
- The coordinate planes are the xy-plane (z = 0), the yz-plane (x = 0) and the zx-plane (y = 0).
- Distance in three dimensions follows from the Pythagorean theorem extended: √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
- The section formula finds a point dividing a segment in a given ratio, using weighted averages of the coordinates.
- Signs of coordinates decide which octant a point lies in, just as signs decide quadrants in two dimensions.
Formulas and facts to remember
- Distance between two pointsIf P = (x₁, y₁, z₁) and Q = (x₂, y₂, z₂), thenPQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].Meaning: extend the 2-D distance formula by adding the square of the difference in z-coordinates.
- Distance from the originOP = √(x² + y² + z²) for the point P = (x, y, z).
- Section formula (internal division)If R divides PQ internally in the ratio m : n, thenR = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)).Meaning: each coordinate of R is the weighted average, with weights m and n.
- Section formula (external division)Replace + with − in denominators and cross-terms:R = ((mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n), (mz₂ − nz₁)/(m − n)), provided m ≠ n.
- MidpointWhen m = n = 1, the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2).
- Centroid of a triangleFor vertices A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃), the centroid isG = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3).
Common mistakes
- Forgetting to square all three differences → always include (z₂ − z₁)² inside the square root.
- Using the wrong sign in external division → remember denominators become (m − n), and the cross-term uses a minus sign.
- Placing a point in the wrong octant because of sign errors → check that each coordinate's sign matches its position relative to the coordinate planes.
- Taking the centroid formula with factor 1/2 instead of 1/3 → the centroid divides each median in ratio 2 : 1 from a vertex, so divide by 3.
- Writing (x, y) instead of (x, y, z) when a third coordinate is zero → always write all three coordinates; if z = 0, write it as 0.
Quick revision
- Three axes, eight octants; every point needs three coordinates (x, y, z).
- Distance formula: √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
- Section formula: weighted averages with m and n; plus for internal, minus for external.
- Midpoint: average each coordinate.
- Centroid: (sum of x)/3, (sum of y)/3, (sum of z)/3.