Match each multi-step movement track in List I with its corresponding net displacement vec… — State PSC Exam

Match each multi-step movement track in List I with its corresponding net displacement vector (shortest distance and direction from the starting point) in List II.

Track I: Starting facing North, a field surveyor turns $45^\circ$ right and walks $12\sqrt{2}\text{ m}$ , then turns $135^\circ$ right and walks $20\text{ m}$ , and finally turns $90^\circ$ left and walks $3\text{ m}$ . $17\text{ m}$ , South-East
Track II: A drone flies $16\text{ m}$ due East, turns $90^\circ$ left and flies $18\text{ m}$ , turns $135^\circ$ left and flies $10\sqrt{2}\text{ m}$ , turns $135^\circ$ right and flies $4\text{ m}$ , and finally turns $90^\circ$ left and flies $6\text{ m}$ . $12\text{ m}$ , North
Track III: At sunrise, a traveler walks $15\text{ m}$ directly towards his shadow, turns $90^\circ$ right and walks $9\text{ m}$ , and then turns $135^\circ$ right and walks $15\sqrt{2}\text{ m}$ . $6\text{ m}$ , South
Track IV: A security officer walks $5\text{ m}$ due South, turns $45^\circ$ left and walks $12\sqrt{2}\text{ m}$ , turns $135^\circ$ left and walks $22\text{ m}$ , and finally turns $90^\circ$ left and walks $7\text{ m}$ . $5\sqrt{2}\text{ m}$ , North-East

Answer

Track I matches with 17 m, South-East; Track II matches with 12 m, North; Track III matches with 6 m, South; Track IV matches with 5√2 m, North-East.

Each movement track is correctly paired with its net displacement vector by projecting sequential movements onto a standard Cartesian coordinate grid: Track I resolves to (15, -8) which is 17 m South-East; Track II resolves to (0, 12) which is 12 m North; Track III resolves to (0, -6) which is 6 m South; Track IV resolves to (5, 5) which is 5√2 m North-East.

Step-by-Step Solution

Decompose Track I using Cartesian coordinates.

Path vector components:

(+12, +12) + (0, -20) + (+3, 0) = (+15, -8)

. Distance =

\sqrt{15^2 + (-8)^2} = 17\text{ m}

, direction is South-East.

Resolving each angular turn into

x

and

y

displacements determines net position accurately.

Decompose Track II using Cartesian coordinates.

Path vector components:

(+16, 0) + (0, +18) + (-10, -10) + (0, +4) + (-6, 0) = (0, +12)

. Distance =

12\text{ m}

, direction is North.

Sequential tracking of direction changes ensures exact final location.

Determine initial cardinal direction for Track III using solar shadow properties, then resolve coordinates.

Sunrise shadow points West. Path vectors:

(-15, 0) + (0, +9) + (+15, -15) = (0, -6)

. Distance =

6\text{ m}

, direction is South.

The morning shadow implicitly fixes the initial movement along the negative x-axis.

Decompose Track IV using Cartesian coordinates.

Path vector components:

(0, -5) + (+12, -12) + (0, +22) + (-7, 0) = (+5, +5)

. Distance =

\sqrt{5^2 + 5^2} = 5\sqrt{2}\text{ m}

, direction is North-East.

Sub-cardinal movement vectors resolved via

\sqrt{2}

multiples simplify to exact integer Cartesian shifts.

Key Concept

Direction and Distance Test - Vector Decomposition and Relative Displacement

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