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Difficulty: MediumScalars and Vectors

A hiker walks 12 km12\text{ km} due East, then 9 km9\text{ km} due South, and finally 4 km4\text{ km} due North. What is the magnitude of the hiker's total displacement from the starting point?

  1. 13 km13\text{ km}Answer
  2. B
    25 km25\text{ km}
  3. C
    15 km15\text{ km}
  4. D
    7 km7\text{ km}

Answer

The magnitude of the hiker's total displacement is 13 km13\text{ km}.
Displacement is a vector quantity requiring directional vector addition. The horizontal Eastward component is 12 km12\text{ km}. The net vertical component along the North-South line is 9 km  South−4 km  North=5 km  South9\text{ km \text{ South}} - 4\text{ km \text{ North}} = 5\text{ km \text{ South}}. Because the East and South directions are perpendicular (90∘90^\circ), the resultant displacement magnitude is 122+52=169=13 km\sqrt{12^2 + 5^2} = \sqrt{169} = 13\text{ km}.

Step-by-Step Solution

1
Set up a coordinate system for horizontal (East-West) and vertical (North-South) components.
East is +x+x direction and North is +y+y direction.
Resolving vectors into orthogonal components allows simple independent summation along each axis.
2
Calculate net displacement along the x-axis and y-axis.
Rx=+12 kmR_x = +12\text{ km} and Ry=−9 km+4 km=−5 kmR_y = -9\text{ km} + 4\text{ km} = -5\text{ km}.
North and South act in opposite directions along the y-axis, so their magnitudes subtract.
3
Calculate the magnitude of the resultant displacement vector using Pythagoras' theorem.
∣R⃗∣=Rx2+Ry2=122+(−5)2=144+25=169=13 km|\vec{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13\text{ km}.
Perpendicular components combine geometrically to give the magnitude of the resultant vector.

Key Concept

Vector resolution and addition of non-collinear displacement vectors.
Estimated Time:1m 30s
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Topics

Scalars and Vectors

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