Direction Sense – Reasoning Ability
For the Madhya Pradesh Police Sub‑Inspector (SI) Examination
1. Simple Explanation
Direction sense refers to the orientation of a geometric object (a line, vector, or angle) – i.e., the way it “points” in space.
- In 2‑D geometry it is the sense in which a directed line AB is drawn from point A to point B.
- In 3‑D problems (rare in reasoning papers) it is the orientation of a vector in space.
- In problems of motion and bearings the direction sense tells whether an object moves clockwise (CW) or counter‑clockwise (CCW) from a reference direction.
Key Idea – When a line or angle is directed, we can talk about the direction sense of that line/angle. It tells us which way we are moving or turning.
2. Key Points
| # | Concept | Description | Formula / Rule |
|---|---|---|---|
| 1 | Directed Line | A line with an arrow from A to B. | Direction vector ( \vec{AB} = (x_B-x_A,; y_B-y_A) ) |
| 2 | Direction Sense of a Line | The sense in which we travel from A to B. | If (AB) is directed, then “to the left/right” of AB is determined by the sign of the cross‑product with a test vector. |
| 3 | Angle Direction Sense | Positive angles are measured counter‑clockwise (CCW) from the positive x‑axis. | ( \theta_{\text{CCW}} = \tan^{-1}!\frac{y}{x} ) (adjusted for quadrant) |
| 4 | Clockwise vs Counter‑Clockwise | In bearings, a turn clockwise reduces the bearing angle, counter‑clockwise increases it. | ( \text{New bearing} = \text{Original bearing} \pm \text{Turn angle} ) (sign depends on direction) |
| 5 | Left / Right of a Directed Line | Point P is to the left of AB if ( (x_P-x_A)(y_B-y_A) - (y_P-y_A)(x_B-x_A) > 0 ); otherwise it is to the right. | Cross‑product sign |
| 6 | Direction Sense in Coordinate Geometry | The slope of a directed line is the same as an undirected line, but the sense tells us whether “up” or “down” is forward. | Slope ( m = \dfrac{y_B-y_A}{x_B-x_A} ) (sign indicates sense) |
| 7 | Direction Sense of a Vector | The unit vector ( \hat{u} = \dfrac{\vec{v}}{ | \vec{v} |
| 8 | Direction Sense in Reasoning Problems | Often used to deduce: • Direction of travel of two moving objects • Relative positions (left/right) • Turn angles in navigation |
Combine dot/cross product + quadrant checks |
3. Worked Example
Problem
A point (P(5, 2)) lies on the directed line (AB) where (A(1,1)) and (B(9,9)).
- Find the direction sense of (AB).
- Determine whether (P) is to the left or right of (AB).
- Give the bearing (in degrees) of the direction from (A) to (B).
Step‑by‑Step Solution
-
Direction Vector of (AB)
[ \vec{AB} = (x_B-x_A,; y_B-y_A) = (9-1,; 9-1) = (8,,8) ] The direction sense is towards increasing (x) and (y); i.e., from bottom‑left to top‑right. -
Left / Right Test
Compute cross‑product of (\vec{AB}) with (\vec{AP}\