AHL 3.14 — Vector Equation of a Line in 2D and 3D

A line in two or three dimensions can be described easily and elegantly using vectors.
The vector equation captures both the starting point and the direction of the line in a single expression.
This method is more powerful and flexible than purely Cartesian equations, especially in three dimensions.

📌 Vector Equation of a Line

The vector equation of a line is:

r = a + λb

Where:
• a is the position vector of a point on the line.
• b is the direction vector (gives direction of travel).
• λ is a parameter (can vary over all real numbers).

As λ changes, the point r moves along the entire line.

📌 Interpretation of a and b

  • a fixes the line to a specific location in space.
  • b determines the direction and steepness of the line.
  • If b = (0, 0, 0), the object does not move → not a line.
  • The magnitude |b| represents the speed if λ is time.

📌 Parametric Form of a Line

Writing the vector equation in coordinates gives the parametric form.
If:

a = (x0, y0, z0) and
b = (l, m, n)

Then the line becomes:


x = x0 + λl
y = y0 + λm
z = z0 + λn

This form clearly shows how each coordinate changes as λ varies.

📌 Cartesian Form of a Line (3D)

Eliminate λ from the parametric equations to obtain:


(x − x0) ÷ l = (y − y0) ÷ m = (z − z0) ÷ n

This expresses the same line but without a parameter.
Useful for comparing two lines or checking intersection/parallelism.

📌 Worked Example

Example: Find the parametric and Cartesian equations of the line passing through
A(2, −1, 3) with direction vector b = (4, 2, −1).

Step 1 — Write vector equation:
a = (2, −1, 3), b = (4, 2, −1)

r = (2, −1, 3) + λ(4, 2, −1)

Step 2 — Parametric:
x = 2 + 4λ
y = −1 + 2λ
z = 3 − λ

Step 3 — Cartesian:
(x − 2) ÷ 4 = (y + 1) ÷ 2 = (z − 3) ÷ (−1)

📌 Angle Between Two Lines

The angle between two lines equals the angle between their direction vectors.
If direction vectors are b and d, then:


cosθ = (b · d) ÷ (|b| × |d|)

• If b · d = 0 → lines are perpendicular.
• If b = k × d → lines are parallel or the same line.

Formula-for-the-angle-between-two-vectors-1024×580.png

📌 Kinematics Interpretation

If λ represents time t, then:
b = velocity vector
|b| = speed

This representation allows modelling of straight-line motion in physics, GPS navigation, and tracking objects in space.

🌍 Real-World Applications

  • GPS uses vector equations for movement tracking and directional guidance.
  • Aircraft flight paths and autonomous navigation are modelled with vector lines.
  • 3D computer graphics use vector lines for camera paths and rendering.
  • Kinematics: modelling constant velocity motion.

📐 IA Opportunities

  • Investigate collisions of moving particles using vector line intersection.
  • Analyse the geometry of 3D camera motion paths.
  • Explore shortest distances from points to lines using vector projection.
  • Study flight path corrections under wind vectors.

🌐 EE Connections

  • How vector equations generalise to lines in higher dimensions.
  • Historical development of analytic geometry (Descartes, Fermat).
  • Use of parametric equations in advanced physics (motion in fields).

❤️ CAS Ideas

  • Run workshops teaching MYP students about lines using interactive 3D models.
  • Help physics students model constant velocity motion with vector equations.
  • Create a real-world mapping project (e.g., tracking motion in sports).

📱 GDC Usage

  • Store vectors a and b using your calculator’s vector menu.
  • Plot parametric equations to visualise 3D lines.
  • Compute the direction vector of a line instantly.
  • Use dot product functions to find the angle between two lines.
  • Use graphing mode to check intersections of lines numerically.

🔍 TOK Perspective

  • Why do mathematicians use different representations for the same line?
  • Does one representation provide “better knowledge” than another?
  • How do symbolic forms affect how we perceive geometric objects?

📝 Examiner Tips

  • Ensure the direction vector is never (0,0,0).
  • When comparing two lines, check direction vectors first (parallel? perpendicular?).
  • For angle problems, always normalise using magnitudes.
  • Write clearly labelled parametric equations — common marks lost here.
  • If one direction vector is a scalar multiple of the other → lines are parallel.