Three Dimensional Geometry
1. Introduction
Three-dimensional geometry extends coordinate geometry to space. It involves points, lines, and planes in 3D, their equations and relationships — a core topic for ISC mathematics.
2. Direction Cosines and Ratios
2.1 Direction Cosines
The direction cosines (l, m, n) of a line are the cosines of the angles α, β, γ that the line makes with the positive x, y, z axes.
l = cos α, m = cos β, n = cos γ. Property: l² + m² + n² = 1.
2.2 Direction Ratios
Any triple (a, b, c) proportional to (l, m, n) is a set of direction ratios.
l = a/√(a² + b² + c²), m = b/√(a² + b² + c²), n = c/√(a² + b² + c²).
3. Equation of a Line
3.1 Vector Form
Through point with position vector a and parallel to vector b: r = a + λb.
3.2 Cartesian Form
Through (x₁, y₁, z₁) with direction ratios (a, b, c): (x - x₁)/a = (y - y₁)/b = (z - z₁)/c = λ.
'If any direction ratio is zero, the corresponding numerator is also zero — not the whole fraction.'
4. Equation of a Plane
4.1 Through Normal Vector
r · n̂ = d (vector form), where n̂ is the unit normal. l x + m y + n z = d (Cartesian form).
4.2 Through Three Points
If three points are given, use determinant form: |x - x₁, y - y₁, z - z₁; x₂ - x₁, y₂ - y₁, z₂ - z₁; x₃ - x₁, y₃ - y₁, z₃ - z₁| = 0.
4.3 Intercept Form
x/a + y/b + z/c = 1, where a, b, c are intercepts on axes.
4.4 Plane Through Line
A plane containing the line r = a + λb and point c: (r - a) · (b × (c - a)) = 0.
5. Angle Between a Line and a Plane
sin θ = |b · n|/(|b||n|), where b is the direction vector of the line and n is the normal to the plane.
6. Distance of a Point from a Plane
Distance from (x₁, y₁, z₁) to plane ax + by + cz + d = 0: D = |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).
7. Skew Lines
Lines that are neither parallel nor intersecting. The shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂ is:
SD = |(a₂ - a₁) · (b₁ × b₂)|/|b₁ × b₂|.
8. Coplanarity of Two Lines
Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar iff:
(a₂ - a₁) · (b₁ × b₂) = 0
This condition implies that the lines either intersect or are parallel.
9. Image of a Point in a Plane
To find the image of point P(x₁, y₁, z₁) in the plane ax + by + cz + d = 0:
- Write the equation of the line through P perpendicular to the plane (direction ratios a, b, c).
- Find the foot of the perpendicular (M) by solving the line and plane.
- The image Q satisfies: M is the midpoint of PQ.
This is a frequently asked problem in ISC examinations.
10. Family of Planes
A plane passing through the line of intersection of two planes: P₁ + λP₂ = 0, where P₁ = a₁x + b₁y + c₁z + d₁ = 0 and P₂ = a₂x + b₂y + c₂z + d₂ = 0.
This family approach is useful when additional conditions (passing through a point, parallel to another plane) are specified.
11. Worked Problems
Problem 1: Find the direction cosines of the line joining A(1, 2, 3) and B(4, 6, 3). Solution: Direction ratios: (3, 4, 0). √(9+16+0) = 5. l = 3/5, m = 4/5, n = 0.
Problem 2: Find the equation of the plane through (2, 3, 1) with normal parallel to (1, 2, 3). Solution: Plane: 1(x-2) + 2(y-3) + 3(z-1) = 0 ⇒ x + 2y + 3z = 11.
Problem 3: Find the angle between the plane x + y + z = 1 and the line (x-1)/2 = (y+1)/1 = (z-3)/-1. Solution: b = (2, 1, -1), n = (1, 1, 1). sin θ = |2+1-1|/(√6 × √3) = 2/√18 = 2/(3√2) = √2/3.
9. Common Mistakes
'Students often confuse the equation of a plane through a given point with a normal vector. Remember: a(x - x₁) + b(y - y₁) + c(z - z₁) = 0, not a x + b y + c z = 0.'
10. ISC Exam Focus
| Topic | Theory Marks | Practical Marks |
|---|---|---|
| Direction cosines and ratios | 2 | 1 |
| Line equations | 3 | 2 |
| Plane equations | 4 | 2 |
| Angle and distance | 4 | 3 |
| Skew lines | 3 | 2 |
11. Self-Test Questions
- Find the angle between the lines (x-1)/2 = (y+2)/-1 = z/3 and x/3 = (y-1)/2 = (z+2)/1.
- Find the distance of (1, 2, 3) from the plane 2x - y + 2z = 10.
- Find the equation of plane through (1, 2, 3), (2, 3, 1), and (3, 1, 2).
- Find the shortest distance between (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-3)/4 = (z-4)/5.
- Show that the lines (x-1)/1 = (y+1)/2 = z/3 and (x-2)/2 = (y-3)/4 = (z-4)/6 are parallel.
