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Zorluk: Çok zorScalars and Vectors

An autonomous drone navigating an obstacle course experiences three mutually perpendicular velocity vectors simultaneously: a horizontal forward velocity of 12 m s−112\text{ m s}^{-1} due East, a horizontal crosswind drift velocity of 9 m s−19\text{ m s}^{-1} due North, and a vertical downdraft velocity of 8 m s−18\text{ m s}^{-1} directed straight downward. What is the magnitude of the resultant velocity vector of the drone in m s−1\text{m s}^{-1}?

Cevap: 17 m s^-1

Cevap

17 m s⁻¹
Because the three velocity components are mutually perpendicular, the magnitude of the overall resultant velocity is calculated using the 3D Pythagorean theorem: 122+92+82=144+81+64=289=17 m s−1\sqrt{12^2 + 9^2 + 8^2} = \sqrt{144 + 81 + 64} = \sqrt{289} = 17\text{ m s}^{-1}.

Adım Adım Çözüm

1
Identify the perpendicular vector components
vx=12 m s−1v_x = 12\text{ m s}^{-1}, vy=9 m s−1v_y = 9\text{ m s}^{-1}, vz=8 m s−1v_z = 8\text{ m s}^{-1}
The three given velocity vectors act along mutually orthogonal spatial axes (East, North, and Downward).
2
Apply the 3D vector resultant magnitude formula
vr=vx2+vy2+vz2v_r = \sqrt{v_x^2 + v_y^2 + v_z^2}
Since the vector components are perpendicular to one another, the magnitude of their resultant is given by the extension of the Pythagorean theorem to three dimensions.
3
Substitute values and evaluate
vr=144+81+64=289=17 m s−1v_r = \sqrt{144 + 81 + 64} = \sqrt{289} = 17\text{ m s}^{-1}
Squaring each component, adding them, and taking the principal square root yields the total magnitude of the velocity vector.

Anahtar Kavram

Magnitude of three mutually perpendicular vector components using the 3D Pythagorean theorem
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Konular

Scalars and Vectors

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