Chapter 1 · Class 11-12 & JEE Main/Adv
3D Geometry
69 topics in 17 parts
Key ideas
| Property | Direction Cosines (l, m, n) | Direction Ratios (a, b, c) |
|---|---|---|
| Definition | Actual cosines of angles made with axes (cos α, cos β, cos γ) | Any set of numbers proportional to direction cosines |
| Normalization | Strictly normalized: l² + m² + n² = 1 | Not normalized; can be multiplied by any constant k ≠ 0 |
| Uniqueness | Unique for a given directed line (up to ± sign) | Infinitely many sets for the same line |
| Unit Vector | lî + mĵ + nk̂ is always a UNIT VECTOR of magnitude 1 | aî + bĵ + ck̂ has magnitude √(a² + b² + c²) |
Step-by-Step Solved Problem (JEE Main Standard)
Problem: A line makes angles of 60° and 45° with the positive X and Y axes respectively. Find the angle it makes with the positive Z axis.
α = 60° ⟹ l = cos 60° = 1/2.
β = 45° ⟹ m = cos 45° = 1/√2.
Let angle with Z axis be γ ⟹ n = cos γ.
l² + m² + n² = 1 ⟹ (1/2)² + (1/√2)² + cos²γ = 1
1/4 + 1/2 + cos²γ = 1 ⟹ 3/4 + cos²γ = 1
cos²γ = 1 - 3/4 = 1/4 ⟹ cos γ = ± 1/2.
If cos γ = 1/2 ⟹ γ = 60° (or π/3).
If cos γ = -1/2 ⟹ γ = 120° (or 2π/3).
Conceptual Foundation & Core Theory
In 3D Euclidean space, a Plane is a flat two-dimensional surface that extends infinitely in all directions. A plane is uniquely positioned in space if we know either (1) its perpendicular distance from the origin and its normal direction, or (2) a point lying on the plane and a normal vector perpendicular to it.
General Cartesian Equation of a Plane:
Every first-degree equation in x, y, z represents a plane:
Ax + By + Cz + D = 0
Crucially, the coefficients <A, B, C> represent the Direction Ratios of the Normal Vector perpendicular to the plane!
Intercept Form: If a plane cuts off intercepts a, b, c from the coordinate axes X, Y, Z respectively, its equation is:
x / a + y / b + z / c = 1
Standard Formulas & Governing Conditions
// Plane Formulations in 3D Space
1. General Cartesian Form:
Ax + By + Cz + D = 0 (Normal vector n = Aî + Bĵ + Ck̂)
2. Vector Form Passing through a with Normal n:
(r - a) · n = 0 ⟺ r · n = d
3. Normal Form (Unit Normal n̂ and Perpendicular Distance p from Origin):
r · n̂ = p ⟺ lx + my + nz = p
4. Intercept Form (Intercepts a, b, c on axes):
x/a + y/b + z/c = 1
5. Perpendicular Distance of Point (x1, y1, z1) from Plane Ax + By + Cz + D = 0:
d = | Ax1 + By1 + Cz1 + D | / √(A² + B² + C²)
6. Distance Between Two Parallel Planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0:
d = | D1 - D2 | / √(A² + B² + C²)Standard Forms of Plane Equations
| Plane Equation Form | Formula | Geometrical Information Conveyed |
|---|---|---|
| General Form | Ax + By + Cz + D = 0 | Normal direction ratios are <A, B, C> |
| Normal Form | lx + my + nz = p | p is exact perpendicular distance from origin; <l, m, n> are DCs |
| Intercept Form | x/a + y/b + z/c = 1 | Plane cuts X at (a,0,0), Y at (0,b,0), Z at (0,0,c) |
| Three-Point Form | | x-x1 y-y1 z-z1 ; x2-x1 y2-y1 z2-z1 ; x3-x1 y3-y1 z3-z1 | = 0 | Plane passing through three non-collinear points |
Step-by-Step Solved Problem (JEE Main Standard)
Problem: Find the perpendicular distance between the parallel planes 2x - 2y + z + 3 = 0 and 4x - 4y + 2z + 5 = 0.
Multiply Plane 1 by 2: 4x - 4y + 2z + 6 = 0.
Plane 2: 4x - 4y + 2z + 5 = 0.
Here A = 4, B = -4, C = 2, D₁ = 6, D₂ = 5.
d = |D₁ - D₂| / √(A² + B² + C²)
d = |6 - 5| / √(4² + (-4)² + 2²) = 1 / √(16 + 16 + 4) = 1 / √36 = 1/6 units.
Analytical Comparison & Distinction Matrix
| Analytical Method | Standard Representation | Parametric / Invariant Framework |
|---|---|---|
| Algebraic Nature | Explicit equations relating spatial coordinates | Parameterized using scalar parameters (λ, μ) or vectors |
| Symmetry Usage | Relies on coordinate expansion | Exploits vector dot/cross products for rapid simplification |
| Computational Efficiency | Direct for intercept evaluation | Superior for angles, orthogonal projections, and distances |
| Common Mistake | Sign confusion in determinants | Omitting boundary condition checks |
Conceptual Foundation & Core Theory
The Vector Equation of a Line in 3D space is the most compact and elegant formulation of linear geometry. It states that the position vector r of any arbitrary point P on the line is equal to the position vector a of a fixed reference point on the line, plus a scalar multiple of a vector b that defines the line's direction:
r = a + λb, λ ∈ ℝ
Geometric Interpretation of the Scalar Parameter λ:
• When λ = 0, r = a, which is the fixed point itself.
• When λ > 0, P lies on one side of point A in the direction of vector b.
• When λ , P lies on the opposite side of point A.
• If b is chosen as a unit vector b̂, then |λ| directly equals the absolute Euclidean distance from point A to point P!
| Vector Form Component | Cartesian Form Equivalent | Physical / Geometrical Meaning |
|---|---|---|
| Position Vector r | xî + yĵ + zk̂ | Coordinates of arbitrary running point P(x, y, z) |
| Base Point Vector a | x₁î + y₁ĵ + z₁k̂ | Coordinates of a known fixed point A(x₁, y₁, z₁) on the line |
| Direction Vector b | aî + bĵ + ck̂ | Direction ratios <a, b, c> governing the line's orientation |
| Scalar Parameter λ | (x - x₁)/a = (y - y₁)/b = (z - z₁)/c | Free scalar parameter determining position along the line |
Step-by-Step Solved Problem (JEE Main Standard)
Problem: Find the angle between the pair of lines given by r = (3î + 2ĵ - 4k̂) + λ(î + 2ĵ + 2k̂) and r = (5î - 2ĵ) + μ(3î + 2ĵ + 6k̂).
b₁ = î + 2ĵ + 2k̂ and b₂ = 3î + 2ĵ + 6k̂.
b₁ · b₂ = (1)(3) + (2)(2) + (2)(6) = 3 + 4 + 12 = 19.
|b₁| = √(1² + 2² + 2²) = √9 = 3.
|b₂| = √(3² + 2² + 6²) = √(9 + 4 + 36) = √49 = 7.
cos θ = |b₁ · b₂| / (|b₁| |b₂|) = 19 / (3 × 7) = 19 / 21.
Angle θ = cos⁻¹(19 / 21) ≈ 25.2°.
Conceptual Foundation & Core Theory
By Euclid's fundamental postulate extended into three dimensions, two distinct points in space uniquely determine a straight line. If point A(x₁, y₁, z₁) and point B(x₂, y₂, z₂) lie on the line, the vector joining them AB = (x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂ provides the line's direction vector.
Cartesian Equation (Two-Point Form):
(x - x₁) / (x₂ - x₁) = (y - y₁) / (y₂ - y₁) = (z - z₁) / (z₂ - z₁) = λ
Notice that the denominators (x₂ - x₁), (y₂ - y₁), (z₂ - z₁) directly represent the direction ratios of the line.
One-Point Form vs Two-Point Form
| Characteristic | Line Through Point & Direction Vector | Line Through Two Points |
|---|---|---|
| Required Data | One point (x1, y1, z1) and direction ratios <a, b, c> | Two coordinates (x1, y1, z1) and (x2, y2, z2) |
| Direction Ratios | Given directly as <a, b, c> | Calculated as <x2 - x1, y2 - y1, z2 - z1> |
| Vector Equation | r = a + λb | r = a + λ(b - a) |
| Midpoint Coordinate | Requires specifying λ = distance/(2|b|) | Direct midpoint formula: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2) |
Step-by-Step Solved Problem (Board Exam Standard)
Problem: Find the Cartesian and vector equations of the line passing through points A(3, 4, -7) and B(1, -1, 6). Determine whether the point C(5, 9, -20) lies on this line.
<a, b, c> = <x₂ - x₁, y₂ - y₁, z₂ - z₁> = <1 - 3, -1 - 4, 6 - (-7)> = <-2, -5, 13>.
• Vector Form: r = (3î + 4ĵ - 7k̂) + λ(-2î - 5ĵ + 13k̂)
• Cartesian Form: (x - 3) / -2 = (y - 4) / -5 = (z + 7) / 13 = λ
Substitute (5, 9, -20) into the Cartesian equation:
(5 - 3) / -2 = 2 / -2 = -1
(9 - 4) / -5 = 5 / -5 = -1
(-20 + 7) / 13 = -13 / 13 = -1
Since λ = -1 for all three coordinates, point C lies on the line (collinear)!
Conceptual Foundation & Core Theory
The orientation of any directed line in 3D space is uniquely characterized by its Direction Angles (α, β, γ)—the angles that the directed line makes with the positive directions of the coordinate axes X, Y, and Z respectively.
Direction Cosines (l, m, n):
The cosines of these direction angles are termed the Direction Cosines (DCs):
l = cos α, m = cos β, n = cos γ
By projecting a position vector r = xî + yĵ + zk̂ onto the axes, we know x = lr, y = mr, z = nr. Since r² = x² + y² + z² = (l² + m² + n²)r², dividing by r² yields the most fundamental identity of 3D geometry:
l² + m² + n² = cos²α + cos²β + cos²γ = 1
And using sin²θ = 1 - cos²θ, we obtain:
sin²α + sin²β + sin²γ = (1 - l²) + (1 - m²) + (1 - n²) = 3 - (l² + m² + n²) = 2
Direction Ratios (a, b, c):
Any three real numbers a, b, c that are directly proportional to the direction cosines l, m, n are called Direction Ratios (DRs):
l/a = m/b = n/c = ± 1 / √(a² + b² + c²)
Direction ratios are not normalized (they can be scaled by any non-zero constant), whereas direction cosines are always normalized unit values (l² + m² + n² = 1).
Standard Formulas & Governing Conditions
// Direction Cosines & Direction Ratios Formulations
1. Fundamental Identity:
l² + m² + n² = 1 ⟺ cos²α + cos²β + cos²γ = 1
sin²α + sin²β + sin²γ = 2
2. Relation between Direction Cosines and Direction Ratios:
l = ± a / √(a² + b² + c²)
m = ± b / √(a² + b² + c²)
n = ± c / √(a² + b² + c²)
3. Angle between two lines with DCs (l1, m1, n1) and (l2, m2, n2):
cos θ = | l1·l2 + m1·m2 + n1·n2 |
• Perpendicular: l1·l2 + m1·m2 + n1·n2 = 0
• Parallel : l1/l2 = m1/m2 = n1/n2Conceptual Foundation & Core Theory
In three-dimensional space, two lines can either be parallel, intersecting, or skew (non-parallel lines that never intersect because they lie in different planes). For two straight lines to be coplanar, they must lie simultaneously on a single flat plane. In that plane, they must either be parallel or intersect at a single unique point.
The Vector Condition: Consider two lines defined in vector form:
Line 1: r = a₁ + λb₁ Line 2: r = a₂ + μb₂
Here, a₁ and a₂ are position vectors of known points on the lines, while b₁ and b₂ are vectors parallel to the lines. The vector connecting point A on Line 1 to point B on Line 2 is (a₂ - a₁). For both lines to reside on the same plane, (a₂ - a₁) must be completely coplanar with b₁ and b₂. Because the cross-product vector (b₁ × b₂) is normal (perpendicular) to the plane containing both lines, the scalar triple product must strictly vanish:
(a₂ - a₁) · (b₁ × b₂) = 0
The Cartesian Determinant Test: Expressing the lines in symmetric Cartesian form:
(x - x₁)/a₁ = (y - y₁)/b₁ = (z - z₁)/c₁ and (x - x₂)/a₂ = (y - y₂)/b₂ = (z - z₂)/c₂
The necessary and sufficient condition for coplanarity is that the 3×3 determinant equals zero:
| (x₂ - x₁) (y₂ - y₁) (z₂ - z₁) |
| a₁ b₁ c₁ | = 0
| a₂ b₂ c₂ |
Standard Formulas & Governing Conditions
// 3D Geometry: Coplanarity & Intersection Governing Equations
1. Vector Condition for Coplanarity:
(a2 - a1) · (b1 × b2) = 0 [Scalar Triple Product = 0]
2. Cartesian Condition for Coplanarity (Determinant Form):
| x2 - x1 y2 - y1 z2 - z1 |
| a1 b1 c1 | = 0
| a2 b2 c2 |
3. Equation of the Plane Containing Both Coplanar Lines:
| x - x1 y - y1 z - z1 |
| a1 b1 c1 | = 0
| a2 b2 c2 |
4. Shortest Distance Between Non-Coplanar (Skew) Lines:
d = | (a2 - a1) · (b1 × b2) | / | b1 × b2 |Relative Spatial Orientations of Two Lines in 3D Space
| Orientation Type | Coplanarity Status | Shortest Distance (d) | Direction Ratios Condition |
|---|---|---|---|
| Intersecting Lines | Always Coplanar (Share 1 plane) | d = 0 (Unique point of contact) | Non-proportional direction ratios (b1 ≠ k b2) |
| Parallel Lines | Always Coplanar (Share 1 plane) | d = |(a2 - a1) × b| / |b| | Direction ratios are strictly proportional: a1/a2 = b1/b2 = c1/c2 |
| Skew Lines | Non-Coplanar (Cannot share a plane) | d = |(a2-a1)·(b1×b2)| / |b1×b2| > 0 | Non-parallel and non-intersecting lines in space |
| Coincident Lines | Coplanar (Identical lines) | d = 0 (Infinite points of contact) | Same direction ratios and share all points |
Step-by-Step Solved Problem (JEE Advanced Standard)
Problem: Show that the lines L₁: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and L₂: (x - 2)/3 = (y - 3)/4 = (z - 4)/5 are coplanar, and find their point of intersection.
(x₁, y₁, z₁) = (1, 2, 3) and (a₁, b₁, c₁) = (2, 3, 4)
(x₂, y₂, z₂) = (2, 3, 4) and (a₂, b₂, c₂) = (3, 4, 5)
| 1 1 1 ; 2 3 4 ; 3 4 5 | = 1(-1) - 1(-2) + 1(-1) = 0.
Determinant = 0 ⟹ Lines are COPLANAR!
Equating coordinates gives P = (-1, -1, -1).
Checkpoint
1. What is the value of sin²α + sin²β + sin²γ where α, β, γ are the direction angles of a line?
2. What are the direction ratios of the normal to the plane 3x - 4y + 5z = 12?
3. What is the Cartesian equation of a line passing through (1, 2, 3) and parallel to the X-axis?
4. If two lines in vector form r = a₁ + λb₁ and r = a₂ + μb₂ are perpendicular, which relation must hold true?
5. What are the direction ratios of the line passing through (2, 3, 5) and (4, -1, 8)?
6. If two non-parallel lines in space have a shortest distance of zero, they are:
All 69 topics in 3D Geometry
Direction Cosines & Direction Ratios
- Direction Cosines
- Direction Ratios
- Coordinate Axes & Coordinate Planes
- Projections on Axes
Equations of Lines in Space
- Cartesian Form
- Vector Form
- Line Through a Point Parallel to a Vector
- Line Through Two Points
- Section Formula
Angle Between Two Lines
- Perpendicular Lines
- Angles Defined by Direction Ratios
Coplanar & Intersecting Lines
- Conditions for Coplanarity & Intersection
- Collinearity & Intersection
- Common Transversals
Foot of Perpendicular & Image
- Foot of Perpendicular From a Point to a Line
- Image of a Point in a Line
- Perpendicular Distance From a Point to a Line
- Foot of Perpendicular From a Point to a Plane
- Image of a Point in a Plane
Shortest Distance Between Two Lines
- Condition for Lines to Intersect
- Distance Between Two Parallel Lines
- Shortest Distance Between Two Non-Coplanar Lines
Plane
- Cartesian Equation of a Plane
- Vector Equation of a Plane
- Normal Form
- Intercept Form
- Through Three Points
- Through a Point & a Line
- Through Two Parallel Lines
- Through a Point Parallel to Two Vectors
- Parallel to a Given Plane
- Pair of Planes
Angle Between Two Planes
- Condition for Parallelism
- Condition for Perpendicularity
- Points on the Same or Opposite Sides of a Plane
Line-Plane Relationships
- Angle Between a Line & a Plane
- Relative Position of a Line & a Plane
- Line of Intersection of Two Planes
Distances From Points & Planes
- Perpendicular Distance of a Point From a Plane
- Distance Between Parallel Planes
Measured Parallel Distance of a Point
- From a Line Measured Parallel to a Plane
- From a Plane Measured Parallel to a Line
- From a Plane Measured Parallel to a Plane
Planes Through the Intersection of Two Planes
- Cartesian Form
- Vector Form
- Family of Planes Through an Intersection
- Rotation About the Line of Intersection
Angle-Bisector Planes
- Equations of Angle Bisectors
- Bisectors of the Acute & Obtuse Angles
- Bisector Containing the Origin
Tetrahedra
- Properties of a Tetrahedron
- Regular Tetrahedron
- Height of a Regular Tetrahedron
- Angles in a Regular Tetrahedron
- Volume of a Tetrahedron
- Volume of Solids
Spheres
- Equation of a Sphere
- Diameter Form of the Equation of a Sphere
Points, Loci, Areas & Projections in 3D
- Coordinates in Space
- Locus in 3D
- Area in 3D
- Projection in 3D
- Optimization in 3D
- Triangle Centres in 3D
Vector Operations in 3D
- Vector Algebra Basics
- Vector & Cross Products
- Scalar Triple Product
- Vector Triple Product
- Vector Angle Bisectors