Questions
3–5 questions per paper
Difficulty
Medium-Hard
Importance
Core — never skip
Overview
Three-dimensional geometry extends Euclidean concepts into space using coordinates, vectors, and planes. It is a high-scoring pillar of the Class 12 mathematics curriculum that tests your ability to visualize intersections and spatial relationships. Mastering the vector and Cartesian forms of lines and planes is essential for solving complex analytical problems efficiently.
Direction Cosines and Ratios
Direction cosines represent the cosines of angles made by a directed line with the coordinate axes. Direction ratios are proportional to direction cosines and are often used as the components of a vector parallel to a line.
- l = cos α, m = cos β, n = cos γ
- l² + m² + n² = 1
- If direction ratios are a, b, c, then l = a/√(a²+b²+c²)
- Relation between direction ratios and direction cosines is l/a = m/b = n/c
Line in Space
A line in 3D space is determined by a point through which it passes and a vector parallel to the line. Students must be comfortable switching between vector equations and Cartesian coordinate equations.
- Vector equation: r = a + λb
- Cartesian equation: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
- Shortest distance between two skew lines: |((a₂-a₁).(b₁×b₂))|/|b₁×b₂|
- Condition for perpendicularity: a₁a₂ + b₁b₂ + c₁c₂ = 0
- Condition for parallelism: a₁/a₂ = b₁/b₂ = c₁/c₂
Plane
Planes in 3D are defined by a normal vector and a fixed point or by three non-collinear points. Understanding the normal vector 'n' is the key to mastering all plane-related problems.
- Normal form: r.n̂ = d
- General form: Ax + By + Cz + D = 0
- Equation through a point: A(x-x₁) + B(y-y₁) + C(z-z₁) = 0
- Intercept form: x/a + y/b + z/c = 1
- Distance of a point from a plane: |Ax₁ + By₁ + Cz₁ + D| / √(A²+B²+C²+D²)
Formula Sheet
l² + m² + n² = 1
r = a + λb
(x-x₁)/a = (y-y₁)/b = (z-z₁)/c
cos θ = |(b₁·b₂)/(|b₁||b₂|)|
r·n = d
Ax + By + Cz + D = 0
d = |Ax₁ + By₁ + Cz₁ + D| / √(A²+B²+C²+D²)
Shortest distance = |(a₂-a₁)·(b₁×b₂)| / |b₁×b₂|
Exam Tip
Always normalize your line equations to the form (x-x₁)/a before identifying direction ratios, as coefficients of x, y, and z must be positive unity.
Common Mistakes
- Confusing direction ratios with direction cosines; remember that direction ratios are not necessarily normalized.
- Forgetting to check if lines are skew or intersecting before attempting to find the shortest distance.
- Sign errors when extracting direction ratios from the Cartesian equation (x-x₁)/a form, especially when coefficients of x, y, or z are not 1.
More Revision Notes
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