Chapter 29 · Class 11-12 & JEE Main/Adv
Straight Lines
43 topics in 20 parts
Key ideas
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
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. If two non-parallel lines in space have a shortest distance of zero, they are:
All 43 topics in Straight Lines
Different Forms of Line
- Point-Slope Form
- Two-Point Form
- Slope-Intercept Form
- Intercept Form
- Normal or Perpendicular Form
- Parallel & Perpendicular Lines Equations
Distance Form of a Line (Parametric Form)
- Parametric Equation of a Line
- Distance of a Point Along a Given Direction
Distance of a Point from a Line
- Perpendicular Distance of Point from Line
- Distance between Two Parallel Lines
Position of Points Relative to a Line
- Points on Same or Opposite Side of a Line
- Points Relative to Angle Bisectors
Concurrency of Three Lines
- Condition for Concurrency of Three Lines
- Intersection & Triangle Non-Formation Conditions
Equations of Bisectors of the Angles between the Lines
- Acute & Obtuse Angle Bisectors
- Angle Bisectors Containing Origin or Point
Family of Straight Lines
- Lines Passing through Intersection of Two Lines
- Fixed Point of Variable Lines
General Equation of Second Degree
- Condition for Pair of Lines Representation
- Discriminant & Delta Condition
Pair of Straight Lines
- Homogeneous Second Degree Equation
- Angle between Pair of Lines
- Angle Bisectors of Pair of Lines
- Lines Through Origin
Image & Foot of Perpendicular
- Foot of Perpendicular
- Image of Point with Respect to Line
- Reflection of Light Rays & Min-Max Problems
Area in Coordinate System
- Area of Triangle
- Area of Polygon & Quadrilateral
Inclination of a Line
- Slope & Angle of Inclination
- Collinear Points & Distance Formula
Perpendicular Bisector
- Equation of Perpendicular Bisector
Homogenization & Combined Equation of Pair of Lines
- Homogenization of Curve Equation
- Lines Joining Origin to Curve Intersections
Triangle Centers & Properties
- Centroid, Incenter & Excenter
- Circumcenter & Orthocenter
- Euler Line & Side Equations
Locus Problems & Transformation of Axes
- Locus of Moving Point
- Polar Coordinates & Shift of Origin
Section formula
- Internal Section,External Section
Angle between two Lines in 2D
- Angle between two Lines
Properties of Quadrilateral
- Rectangle,Parallelogram,Rhombus,Square,Trapezoid,Kite
Inresection of Two Lines
- Point of Intersection,coincident condition