Match each person's multi-turn displacement sequence in List I with their corresponding fi… — State PSC Exam

Match each person's multi-turn displacement sequence in List I with their corresponding final position relative to their starting point in List II.

Person P: Walks $10\text{ km}$ North, turns $90^\circ$ right and walks $12\text{ km}$ East, then turns $135^\circ$ right and walks $7\sqrt{2}\text{ km}$ South-West. $5\text{ km}$ East and $3\text{ km}$ North
Person Q: Walks $14\text{ km}$ South, turns $90^\circ$ right and walks $5\text{ km}$ West, then turns $135^\circ$ right and walks $9\sqrt{2}\text{ km}$ North-East. $4\text{ km}$ East and $5\text{ km}$ South
Person R: Walks $3\text{ km}$ East, turns $90^\circ$ left and walks $16\text{ km}$ North, then turns $135^\circ$ left and walks $8\sqrt{2}\text{ km}$ South-West. $5\text{ km}$ West and $8\text{ km}$ North
Person S: Walks $11\text{ km}$ West, turns $90^\circ$ left and walks $4\text{ km}$ South, then turns $135^\circ$ left and walks $6\sqrt{2}\text{ km}$ North-East. $5\text{ km}$ West and $2\text{ km}$ North

Person P matches with

5\text{ km}

East and

3\text{ km}

North; Person Q matches with

4\text{ km}

East and

5\text{ km}

South; Person R matches with

5\text{ km}

West and

8\text{ km}

North; Person S matches with

5\text{ km}

West and

2\text{ km}

North.

Each trajectory is resolved correctly by tracking facing direction changes and vector sum components. Person P resolves to

(+5, +3)

, Person Q resolves to

(+4, -5)

, Person R resolves to

(-5, +8)

, and Person S resolves to

(-5, +2)

Set up a Cartesian coordinate system with starting points at

(0,0)

, where North is

+y

, South is

-y

, East is

+x

, and West is

-x

Displacements along cardinal directions align directly with coordinate axes.

Converting directional movements to Cartesian vectors simplifies multi-turn trajectory calculations.

Decompose angular diagonal movements into horizontal and vertical components using

d\sqrt{2}

vector resolution.

A distance of

d\sqrt{2}\text{ km}

in sub-cardinal directions resolves to

|\Delta x| = d

and

|\Delta y| = d

In 45-degree sub-cardinal directions (NE, NW, SE, SW),

\cos(45^\circ) = \sin(45^\circ) = 1/\sqrt{2}

Calculate final coordinates for each person by summing intermediate vector components.

P reaches

(5,3)

; Q reaches

(4,-5)

; R reaches

(-5,8)

; S reaches

(-5,2)

Sequential vector addition yields the accurate net displacement relative to the origin.

Vector resolution of multi-turn cardinal and sub-cardinal displacements

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