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

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 \( F_{motion} \approx 9.16 \).