Direction Sense – Advanced Study Notes
(Reasoning Ability – Meghalaya Police Constable, MIT‑level Treatment)
1. What is “Direction Sense”?
| Aspect | Definition |
|---|---|
| Direction | A unit vector that indicates orientation in 2‑ or 3‑dimensional space. |
| Sense | The polarity of that direction (i.e., “forward” vs. “backward”). |
| Why it matters | Reasoning questions often ask you to infer relative positions, rotate viewpoints, or determine the impact of a movement on a system. |
Key Takeaway: Direction is the line of travel; sense is which end of the line you are moving towards.
2. Coordinate Systems & Direction Vectors
| System | Notation | Direction Vector | Typical Use |
|---|---|---|---|
| Cartesian (2‑D) | ( (x, y) ) | ( \mathbf{v} = \langle v_x, v_y \rangle ) | Grid‑based reasoning |
| Polar (2‑D) | ( (r, \theta) ) | ( \mathbf{v} = r(\cos\theta, \sin\theta) ) | Compass bearings |
| Cartesian (3‑D) | ( (x, y, z) ) | ( \mathbf{v} = \langle v_x, v_y, v_z \rangle ) | 3‑D spatial reasoning |
| Spherical (3‑D) | ( (r, \theta, \phi) ) | ( \mathbf{v} = r(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta) ) | Earth‑centric navigation |
Note: In all cases, the direction is independent of the magnitude ( r ). We usually normalise to unit length: ( \hat{\mathbf{v}} = \mathbf{v}/|\mathbf{v}| ).
3. Fundamental Properties
-
Opposite Direction
[ \hat{\mathbf{v}} ;; \text{and} ;; -\hat{\mathbf{v}} ] represent the same line but opposite senses. -
Orthogonality
[ \hat{\mathbf{u}}\cdot\hat{\mathbf{v}} = 0 ;; \Longrightarrow ;; \hat{\mathbf{u}} \perp \hat{\mathbf{v}} ] Perpendicular directions never interfere in linear superposition. -
Parallelism
[ \hat{\mathbf{u}} = \pm \hat{\mathbf{v}} ;; \Longrightarrow ;; \text{same line} ] -
Angle Between Directions
[ \cos\theta = \hat{\mathbf{u}}\cdot\hat{\mathbf{v}} ] Determines the bearing between two directions. -
Direction of a Sum
If ( \mathbf{u} ) and ( \mathbf{v} ) are non‑zero, the direction of ( \mathbf{u} + \mathbf{v} ) is given by the normalized sum.
Edge case: If ( \mathbf{u} = -\mathbf{v} ), the sum is zero; the direction is undefined.
4. Direction Sense in the Plane (2‑D)
4.1. Compass Bearings
| Bearing | Angle θ (from East) | Direction Vector |
|---|---|---|
| N | ( 90^\circ ) | ( (0, 1) ) |
| NE | ( 45^\circ ) | ( (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) ) |
| E | ( 0^\circ ) | ( (1, 0) ) |
| SE | ( -45^\circ ) | ( (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) ) |
| S | ( -90^\circ ) | ( (0, -1) ) |
| SW | ( -135^\circ ) | ( (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) ) |
| W | ( 180^\circ ) | ( (-1, 0) ) |
| NW | ( 135^\circ ) | ( (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) ) |
Convention: Angles measured counter‑clockwise from the +x (East) axis.
4.2. Relative Direction Problems
-
Rotation:
Rotating a direction by ( \alpha ) degrees:
[ \hat{\mathbf{v}}' = \begin{bmatrix} \cos \alpha & -\sin \alpha\ \sin \alpha & \cos \alpha \end{bmatrix} \hat{\mathbf{v}} ] -
Mirroring:
Mirror across the x‑axis: ( (v_x, v_y) \rightarrow (v_x, -v_y) ).
Mirror across the y‑axis: ( (v_x, v_y) \rightarrow (-v_x, v_y) ). -
Shortest Path:
For two points ( P_1, P_2 ), the direction from ( P_1 ) to ( P_2 ) is
[ \hat{\mathbf{d}} = \frac{P_2 - P_1}{|P_2 - P_1|} ]