Direction sense

Direction Sense – Advanced Study Notes

(Designed for the Chandigarh Patwari / Lekhpal Reasoning Ability course – MIT‑level depth)

1. Executive Summary

Direction sense problems test one’s ability to reason about spatial relations, relative motion, and the effect of rotations. They are ubiquitous in aptitude tests (e.g., “A walks north‑east …”) and require a firm grasp of:

1. Fundamental geometry (angles, bearings, vector addition).
2. Coordinate‑system reasoning (Cartesian & polar).
3. Transformation theory (rotations, reflections, translations).
4. Edge‑case detection (e.g., “exactly opposite”, “colinear”, “zero displacement”).

The following notes present a rigorous, first‑principles treatment, followed by advanced theory (vector‑based proofs, spherical geometry), common pitfalls, and a challenging problem with full solution.

2. Fundamental Concepts

2.1. Compass Directions & Bearings

• Bearing: θ measured clockwise from North.
• Cardinal direction: multiples of 90°.
• Intercardinal: multiples of 45°.

2.2. Coordinate System

• Standard Cartesian 2‑D:
  • X‑axis: East (+), West (–).
  • Y‑axis: North (+), South (–).
• Polar Coordinates:
  • r = distance from origin.
  • θ = bearing (same as above).

2.3. Directional Vectors

For a unit step in direction θ: [ \mathbf{u}(\theta) = (\sin\theta, \cos\theta) ] (Using θ in radians, with θ measured clockwise from North; if measured counter‑clockwise, swap sine/cosine signs accordingly.)

Properties

• Additivity: u(θ1) + u(θ2) gives resultant direction.
• Rotation: ( R_{\phi}(\mathbf{v}) = \mathbf{v} \begin{pmatrix}\cos\phi & -\sin\phi\ \sin\phi & \cos\phi\end{pmatrix} ).
• Reflection over a line at angle α: ( \mathbf{v}' = \mathbf{v} - 2(\mathbf{v}\cdot \mathbf{n})\mathbf{n} ), where n is unit normal.

3. From First Principles – Geometry of Direction

3.1. Relative Direction

Given two points P and Q, the direction from P to Q is defined by the vector
[ \mathbf{d}_{P\to Q} = Q - P. ] The bearing θ satisfies: [ \tan\theta = \frac{\Delta x}{\Delta y} = \frac{Q_x - P_x}{Q_y - P_y}, ] with care taken for quadrant determination.

3.2. Angle Between Directions

Given two bearings θ1 and θ2, the smallest angle between them is: [ \Delta\theta = \min{|\theta_1 - \theta_2|,, 360° - |\theta_1 - \theta_2|}. ] If Δθ = 180°, the directions are opposite; if Δθ = 0°, they are identical.

3.3. Path and Displacement

For a sequence of moves ({ (d_i, \theta_i) }) (distance d and bearing θ), the net displacement vector: [ \mathbf{D} = \sum_{i} d_i,\mathbf{u}(\theta_i). ] The net bearing is the bearing of D.

• If D = (0,0), the traveller has returned to the origin.
• If D is colinear with one of the steps, the path is a straight line.

4. Advanced Theory

4.1. Vector Algebra & Rotations

Let v be a displacement vector. Rotating v by angle φ clockwise: [ v' = R_{\phi}v, ] where ( R_{\phi} ) is the rotation matrix above.
Proof:
For v = (x, y), applying ( R_{\phi} ) gives: [ x' = x\cos\phi + y\sin\phi, \quad y' = -x\sin\phi + y\cos\phi, ] which is exactly the coordinate transformation for clockwise rotation.

Edge case: φ = 0° → identity; φ = 180° → negation of vector.

4.2. Spherical Geometry (Optional)

For long‑distance navigation (e.g., flights), bearings are along great circles on a sphere.

• Bearing formula (from point A to B): [ \theta = \operatorname{atan2}!\bigl(\sin\Delta\lambda \cos\phi_2