Schooling · Class 12 · Mathematics · Three Dimensional Geometry · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-II
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
- A line in space is fixed by a point…
- Direction cosines satisfy l² + m² +…
- Two lines in space are either…
- A plane in space is fixed by a point…
- The angle between two lines equals…
- The shortest distance between skew…
- Formulas and facts
- Direction cosines relation
- Equation of a line through point…
- Vector form of a line
- Equation of a line through two…
- Angle θ between two lines with…
- General equation of a plane
- Equation of plane through (x₁, y₁…
- Watch out for
- Forgetting to take the absolute…
- Confusing direction ratios with…
- Using the wrong normal vector for a…
- Sign errors when subtracting…
- Applying the skew-line distance…
What this chapter is about
Three dimensional geometry extends the coordinate methods you learnt for the plane into the space around you. Instead of two axes (x and y), you now work with three mutually perpendicular axes (x, y and z). Every point in space has an ordered triple (x, y, z), and you can describe lines and planes using algebra.
Key ideas
- A line in space is fixed by a point it passes through and a direction; direction ratios (a, b, c) or direction cosines (l, m, n) describe that direction.
- Direction cosines satisfy l² + m² + n² = 1; direction ratios are any set proportional to the direction cosines.
- Two lines in space are either parallel, intersecting, or skew (neither parallel nor meeting).
- A plane in space is fixed by a point and a normal vector, or by three non-collinear points.
- The angle between two lines equals the angle between their direction vectors; the angle between two planes equals the angle between their normals.
- The shortest distance between skew lines is measured along the common perpendicular to both.
Formulas and facts to remember
- Direction cosines relation: l² + m² + n² = 1, where l = cos α, m = cos β, n = cos γ are the cosines of angles the line makes with the x, y, z axes.
- Equation of a line through point (x₁, y₁, z₁) with direction ratios a, b, c (Cartesian form):(x − x₁)/a = (y − y₁)/b = (z − z₁)/c
- Vector form of a line: r = a + λb, where a is the position vector of a point on the line and b is the direction vector.
- Equation of a line through two points (x₁, y₁, z₁) and (x₂, y₂, z₂):(x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁) = (z − z₁)/(z₂ − z₁)
- Angle θ between two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂):cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁² + b₁² + c₁²) × √(a₂² + b₂² + c₂²)]
- General equation of a plane: ax + by + cz + d = 0, where (a, b, c) is normal to the plane.
- Equation of plane through (x₁, y₁, z₁) with normal (a, b, c):a(x − x₁) + b(y − y₁) + c(z − z₁) = 0
- Distance from point (x₁, y₁, z₁) to plane ax + by + cz + d = 0:Distance = |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)
- Shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂:Distance = |(a₂ − a₁) · (b₁ × b₂)| / |b₁ × b₂|
Common mistakes
- Forgetting to take the absolute value when finding angle or distance → the formulas use modulus to ensure a non-negative result.
- Confusing direction ratios with direction cosines → direction cosines must satisfy l² + m² + n² = 1; direction ratios need not.
- Using the wrong normal vector for a plane → in ax + by + cz + d = 0, the normal is (a, b, c), not (a, b, d).
- Sign errors when subtracting coordinates in the two-point form of a line → keep (x₂ − x₁), (y₂ − y₁), (z₂ − z₁) in order.
- Applying the skew-line distance formula to parallel lines → for parallel lines, the cross product b₁ × b₂ is zero, so a different method (point-to-line distance) is needed.
Quick revision
- Direction cosines: l² + m² + n² = 1; direction ratios are proportional to them.
- Line through (x₁, y₁, z₁) with direction (a, b, c): (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
- Plane with normal (a, b, c): ax + by + cz + d = 0.
- Angle between planes = angle between their normals.
- Distance from point to plane = |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²).
- Skew-line shortest distance uses the scalar triple product divided by |b₁ × b₂|.