📚 Direction Sense – Advanced Study Notes
Exam: Delhi Police Constable (Reasoning Ability)
Level: MIT‑style – first principles to edge cases
Format: Markdown (text + LaTeX)
1. What is “Direction Sense”?
| Concept | Definition | Key Point |
|---|---|---|
| Direction | The line along which an object moves or an observer looks. | Described by a vector or an angle. |
| Bearing | The angular measurement from a reference (usually North) to a line of sight. | Usually expressed in degrees, minutes, seconds. |
| Azimuth | The angle measured clockwise from the North (or another fixed direction) to a point on the horizon. | Same as bearing in most navigation texts. |
| Compass Quadrants | N‑E, E‑S, S‑W, W‑N; each covers 90°. | Helps translate cardinal directions to numerical bearings. |
2. From First Principles
2.1 Vectors & Unit Vectors
A 2‑D direction can be represented as a vector ( \mathbf{v} = (x, y) ).
- Magnitude: ( |\mathbf{v}| = \sqrt{x^2 + y^2} )
- Unit vector: ( \hat{\mathbf{u}} = \frac{\mathbf{v}}{|\mathbf{v}|} )
2.2 Polar Coordinates
( \mathbf{v} = r(\cos\theta, \sin\theta) )
- ( r ) = distance (magnitude)
- ( \theta ) = angle from the positive (x)-axis (East) counter‑clockwise
Bearing is ( \beta = 90^\circ - \theta ) if ( \theta ) is measured from East.
2.3 Direction Cosines (3‑D)
For a 3‑D vector ( \mathbf{v} = (x, y, z) ):
[ \begin{aligned} \alpha &= \cos^{-1}\left(\frac{x}{|\mathbf{v}|}\right)\ \beta &= \cos^{-1}\left(\frac{y}{|\mathbf{v}|}\right)\ \gamma &= \cos^{-1}\left(\frac{z}{|\mathbf{v}|}\right) \end{aligned} ]
These are the angles between the vector and the coordinate axes.
3. Coordinate Transformations
3.1 Rotations
Given a vector ( \mathbf{v} = (x, y) ) and a rotation by angle ( \phi ) (counter‑clockwise):
[ \begin{bmatrix} x'\ y' \end{bmatrix}
\begin{bmatrix} \cos\phi & -\sin\phi\ \sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix} x\ y \end{bmatrix} ]
- Bearing change: ( \beta' = \beta + \phi ) (mod 360°)
- Edge case: If ( \phi = 180^\circ ), the direction reverses; if ( \phi = 0^\circ ), no change.
3.2 Translational Shifts
When moving from point ( A(x_1, y_1) ) to ( B(x_2, y_2) ):
[ \mathbf{AB} = \big( x_2 - x_1,; y_2 - y_1 \big) ]
The bearing from ( A ) to ( B ) is:
[ \beta = \tan^{-1}!\left( \frac{y_2 - y_1}{x_2 - x_1} \right) ]
Note: Use the correct quadrant by adding 180° when ( x_2 < x_1 ).
4. Edge Cases & Pitfalls
| Situation | Observation | How to Handle |
|---|---|---|
| Same point (A = B) | Direction undefined | Flag as “No direction” or “Zero vector.” |
| Collinear points | Bearing may be 0° or 180° | Ensure the sign of the vector components dictates the exact bearing. |
| Angle 90° or 270° | Perpendicular to axes | Avoid division by zero in tangent formulas by using atan2(y, x) function. |
| Wrapping around 360° | 350° + 20° = 10° (not 370°) | Always take modulo 360° after addition/subtraction. |
| Negative coordinates | Bearing may fall in 2nd or 3rd quadrant | Use atan2 to get correct quadrant; otherwise add 180° manually. |
| Compass vs. Azimuth vs. Bearing | Some texts use 0° = North, others 0° = East | Clarify the convention before solving. |
5. Deeper Theory
5.1 Geometric Interpretation
- Direction as a line: In geometry, a line is defined by two points. The direction is given by the slope ( m = \frac{y_2-y_1}{x_2-x_1} ).
- Slope–angle relation: ( \theta = \tan^{-1}(m) ). When ( m ) is undefined (vertical line), the direction is 90° or 270°.
5.2 Vector Dot Product & Direction
The dot product gives the cosine of the angle between two vectors:
[ \mathbf{