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Zorluk: KolayScalars and Vectors

A car moves 80 km80\text{ km} due East along a straight highway and then turns to travel 60 km60\text{ km} due North. What is the magnitude of the displacement of the car from its starting position?

  1. 100 km100\text{ km}Cevap
  2. B
    140 km140\text{ km}
  3. C
    20 km20\text{ km}
  4. D
    70 km70\text{ km}

Cevap

The magnitude of the displacement is 100 km100\text{ km}.
Because displacement is a vector quantity, perpendicular components must be combined vectorially using the Pythagorean theorem rather than scalar addition. The calculation 802+602=100 km\sqrt{80^2 + 60^2} = 100\text{ km} correctly yields the magnitude of the net displacement vector.

Adım Adım Çözüm

1
Identify the vector components and their directions.
Eastward displacement x=80 kmx = 80\text{ km}, Northward displacement y=60 kmy = 60\text{ km}. The directions are mutually perpendicular (90∘90^\circ to each other).
Displacement is a vector quantity, so direction must be accounted for when combining components.
2
Apply the Pythagorean theorem to calculate the resultant vector magnitude.
R=x2+y2=802+602=6400+3600=10000=100 kmR = \sqrt{x^2 + y^2} = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100\text{ km}.
When two vector components act at right angles (90∘90^\circ), the magnitude of their resultant is given by the hypotenuse of the right-angled triangle formed by the vectors.

Anahtar Kavram

Vector Addition of Perpendicular Components
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Konular

Scalars and Vectors

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