An officer leaves his office at point O and walks 12\text{ km} due East to reach point P_1… — State PSC Exam

An officer leaves his office at point $O$ and walks $12\text{ km}$ due East to reach point $P_1$ . He then turns $90^\circ$ to his left and walks $9\text{ km}$ due North to reach point $P_2$ . From $P_2$ , he turns $135^\circ$ in the clockwise direction and walks $5\sqrt{2}\text{ km}$ straight to reach point $P_3$ . Finally, he turns due West and walks $17\text{ km}$ to reach point $P_4$ . What is the shortest distance between his final position $P_4$ and his starting point $O$ , and in which direction is he relative to the starting point?

$4\text{ km}$ , NorthAnswer
B
$4\text{ km}$ , South
C
$5\text{ km}$ , North-East
D
$17\text{ km}$ , West

Answer

4\text{ km}

due North of the starting point.

By resolving each movement step into Cartesian coordinates starting from origin

(0,0)

:
-

P_1 = (12, 0)

P_2 = (12, 9)

- Turning

135^\circ

clockwise from North points South-East. Moving

5\sqrt{2}\text{ km}

gives

\Delta x = +5

and

\Delta y = -5

, so

P_3 = (17, 4)

.
- Moving

17\text{ km}

West gives

P_4 = (0, 4)

.
The shortest distance from

(0,0)

(0,4)

4\text{ km}

due North.

Step-by-Step Solution

Set origin

O

(0, 0)

and calculate coordinates after the first two movements.

After walking

12\text{ km}

East,

P_1 = (12, 0)

. After turning

90^\circ

left (facing North) and walking

9\text{ km}

P_2 = (12, 9)

East represents the

+x

direction and North represents the

+y

direction.

Determine facing direction at

P_2

and calculate displacement for the

135^\circ

clockwise turn.

Facing North (

0^\circ

). A

135^\circ

clockwise turn points South-East. Moving

5\sqrt{2}\text{ km}

South-East gives

\Delta x = 5\sqrt{2} \cdot \cos(45^\circ) = 5\text{ km}

and

\Delta y = -5\sqrt{2} \cdot \sin(45^\circ) = -5\text{ km}

. Thus,

P_3 = (12 + 5, 9 - 5) = (17, 4)

South-East movement increases the

x

-coordinate by

5

and decreases the

y

-coordinate by

5

Calculate final position

P_4

after walking

17\text{ km}

West.

Walking

17\text{ km}

West (

-x

direction) from

(17, 4)

gives

P_4 = (17 - 17, 4) = (0, 4)

Westward displacement directly reduces the

x

-coordinate by

17\text{ km}

Compute shortest distance and direction from origin

(0,0)

(0,4)

Shortest distance

= \sqrt{(0-0)^2 + (4-0)^2} = 4\text{ km}

in the positive

y

-direction (North).

Since the

x

-coordinate is

0

and

y = 4

, the point lies directly on the North axis.

Key Concept

Vector resolution of multi-step directional movements involving angular turns

Estimated Time:1m 30s

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