Straight lines

Straight Lines

MIT‑Level Study Notes (KCET – Mathematics)

Goal – Build a complete understanding of straight lines, starting from Euclid’s axioms and ending with contemporary algebraic, geometric, and analytic tools.
Prerequisites – Familiarity with real analysis, linear algebra, basic topology, and elementary geometry.

1. First‑Principles Definition

1.1 Euclidean Axioms (Reformulated)

1. Postulate 1 – Through any two distinct points there exists exactly one line.
2. Postulate 2 – A line can be extended indefinitely in both directions.
3. Postulate 3 – Given a line and a point not on it, there is exactly one line through the point parallel to the first.

These axioms implicitly define a 1‑dimensional linear manifold—a straight line.

1.2 Formalization via Set Theory

Let ( \mathbb{R}^n ) be the Euclidean space. A line ( L \subset \mathbb{R}^n ) is a set of the form
[ L = {, \mathbf{p} + t\mathbf{v} \mid t \in \mathbb{R} ,}, ] where ( \mathbf{p} \in \mathbb{R}^n ) is a point (the anchor) and ( \mathbf{v} \in \mathbb{R}^n \setminus {\mathbf{0}} ) is a non‑zero vector (the direction).
The pair ((\mathbf{p},\mathbf{v})) is not unique: any ( \mathbf{p}' = \mathbf{p} + s\mathbf{v}) yields the same line.

2. Algebraic Forms

2.1 Cartesian Equation in ( \mathbb{R}^2 )

Given two distinct points ( P=(x_1,y_1), Q=(x_2,y_2) ):

1. Slope ( m = \dfrac{y_2-y_1}{x_2-x_1} ) (undefined if (x_2=x_1)).
2. Intercept ( c = y_1 - m x_1 ).
3. Equation ( y = mx + c ) (vertical line: ( x = x_1 )).

2.2 Vector (Normal) Equation in ( \mathbb{R}^3 )

Let ( \mathbf{n} ) be a normal vector. The line is [ \mathbf{n} \cdot (\mathbf{x} - \mathbf{p}) = 0. ] In coordinates:
[ a(x-x_0) + b(y-y_0) + c(z-z_0) = 0, ] with ( \mathbf{n}=(a,b,c)\neq \mathbf{0} ).

3. Geometry in the Plane (( \mathbb{R}^2 ))

3.1 Distance from a Point to a Line

Given ( L: y = mx + c ) and point ( P=(x_0,y_0) ): [ d(P,L) = \frac{|mx_0 - y_0 + c|}{\sqrt{m^2+1}}. ] If ( L ) is vertical ( (x = x_0) ), then ( d = |x_0 - x_1| ).

3.2 Intersection of Two Lines

Let
( L_1: a_1x + b_1y = c_1 ),
( L_2: a_2x + b_2y = c_2 ).

Solve linear system: [ \Delta = a_1b_2 - a_2b_1. ]

• If ( \Delta \neq 0 ), unique intersection ( P ).
• If ( \Delta = 0 ) and ( (a_1,b_1,c_1) ) proportional to ( (a_2,b_2,c_2) ), lines coincide.
• Otherwise, lines are parallel and disjoint.

3.3 Parallel & Perpendicular Lines

• Parallel: ( m_1 = m_2 ). In vector form, direction vectors are scalar multiples.
• Perpendicular: ( m_1 m_2 = -1 ). In vector form, dot product ( \mathbf{v}_1 \cdot \mathbf{v}_2 = 0 ).

3.4 Edge Cases

4. Geometry in Three Dimensions (( \mathbb{R}^3 ))

4.1 Parametric Form

[ \mathbf{x}(t) = \mathbf{p} + t\mathbf{v}, \quad