A hiker walks 12\text{ km} due East, then 9\text{ km} due South, and finally 4\text{ km} d… — JAMB UTME

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

$13\text{ km}$ Cevap
B
$25\text{ km}$
C
$15\text{ km}$
D
$7\text{ km}$

Cevap

The magnitude of the hiker's total displacement is

13\text{ km}

Displacement is a vector quantity requiring directional vector addition. The horizontal Eastward component is

12\text{ km}

. The net vertical component along the North-South line is

9\text{ km \text{ South}} - 4\text{ km \text{ North}} = 5\text{ km \text{ South}}

. Because the East and South directions are perpendicular (

90^\circ

), the resultant displacement magnitude is

\sqrt{12^2 + 5^2} = \sqrt{169} = 13\text{ km}

Adım Adım Çözüm

Set up a coordinate system for horizontal (East-West) and vertical (North-South) components.

East is

+x

direction and North is

+y

direction.

Resolving vectors into orthogonal components allows simple independent summation along each axis.

Calculate net displacement along the x-axis and y-axis.

R_x = +12\text{ km}

and

R_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.

Calculate the magnitude of the resultant displacement vector using Pythagoras' theorem.

|\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.

Anahtar Kavram

Vector resolution and addition of non-collinear displacement vectors.

Tahmini Süre:1m 30s

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