Question

What is the y-component and magnitude of a vector 𝑎 located in the xy-plane if its direction is 230° counterclockwise rotation movement measured from positive x and if its x component is - 6.5 m?

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Answer to a math question What is the y-component and magnitude of a vector 𝑎 located in the xy-plane if its direction is 230° counterclockwise rotation movement measured from positive x and if its x component is - 6.5 m?

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Jayne
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73 Answers
Para encontrar la componente y y la magnitud del vector 𝑎, primero debemos descomponer el vector en sus componentes x e y utilizando la información dada sobre la dirección del vector y su componente x.

Dado que la dirección del vector es 230° en sentido contrario al movimiento de giro de las manecillas del reloj medido desde el eje x positivo, podemos determinar la componente x y la componente y utilizando trigonometría.

La componente x del vector se puede encontrar utilizando la función coseno, y la componente y se puede encontrar utilizando la función seno, como sigue:

Componente x:
a_x = |a| \cdot \cos(\theta)
a_x = |-6.5| \cdot \cos(230^\circ)
a_x = -6.5 \cdot \cos(230^\circ)

Componente y:
a_y = |a| \cdot \sin(\theta)
a_y = |-6.5| \cdot \sin(230^\circ)
a_y = -6.5 \cdot \sin(230^\circ)

Ahora podemos calcular las componentes y la magnitud del vector:

a_x = -6.5 \cdot \cos(230^\circ) \approx -6.5 \cdot 0.7660 \approx -4.971\, m
a_y = -6.5 \cdot \sin(230^\circ) \approx -6.5 \cdot (-0.6428) \approx 4.178\, m

La magnitud del vector 𝑎 se puede calcular utilizando el teorema de Pitágoras:
|a| = \sqrt{a_x^2 + a_y^2}
|a| = \sqrt{(-4.971)^2 + (4.178)^2}
|a| = \sqrt{24.71 + 17.43}
|a| \approx \sqrt{42.14}
|a| \approx 6.49\, m

Por lo tanto, la componente y del vector es 4.178 m y la magnitud del vector es 6.49 m.

\textbf{Respuesta:} La componente y del vector es 4.178 m y la magnitud del vector es 6.49 m.

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