A force \( \mathbf{F}=[7,6] \) is applied to move a car in the direction \( \mathbf{d}=[-8,-9] \). What is the magnitude of the force in the direction of motion? magnitude of \( \mathbf{F}= \) Ex. 1.2
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
Extra Insights
To find the magnitude of the force in the direction of motion, we first need to find the unit vector of the direction. The magnitude of the direction vector \( \mathbf{d} = [-8, -9] \) is calculated as \( \sqrt{(-8)^2 + (-9)^2} = \sqrt{64 + 81} = \sqrt{145} \). The unit vector \( \mathbf{u_d} \) in the direction of \( \mathbf{d} \) is then \( \mathbf{u_d} = \frac{1}{\sqrt{145}} [-8, -9] = \left[-\frac{8}{\sqrt{145}}, -\frac{9}{\sqrt{145}}\right] \). The magnitude of the force in the direction of motion can be found using the dot product: \[ F_{motion} = \mathbf{F} \cdot \mathbf{u_d} = [7, 6] \cdot \left[-\frac{8}{\sqrt{145}}, -\frac{9}{\sqrt{145}}\right] = -\frac{56}{\sqrt{145}} - \frac{54}{\sqrt{145}} = -\frac{110}{\sqrt{145}}. \] The magnitude we are looking for is the absolute value \( \left| F_{motion} \right| = \frac{110}{\sqrt{145}} \). So the final answer for the magnitude of the force in the direction of motion is approximately \( F_{motion} \approx 9.16 \).
