A force \( \mathbf{F}=[0,1,-2] \) is applied to move a particle that is constrained to move in the direction \( \mathbf{d}=[7,-9,2] \). What is the magnitude of the force in the direction of motion? magnitude of \( \mathbf{F}= \) Ex: 1.2 In which direction does the particle move? \( \checkmark \) Select Along -d, opposite to the direction of motion Along d, in the direction of motion

To find the magnitude of the force in the direction of motion, we first need to project the force vector \( \mathbf{F} \) onto the direction vector \( \mathbf{d} \). The unit vector of \( \mathbf{d} \) is given by: \[ \mathbf{u_d} = \frac{\mathbf{d}}{|\mathbf{d}|} \] where \( |\mathbf{d}| = \sqrt{7^2 + (-9)^2 + 2^2} = \sqrt{49 + 81 + 4} = \sqrt{134} \). Next, the projection of \( \mathbf{F} \) onto \( \mathbf{d} \) is calculated as: \[ \text{proj}_{\mathbf{d}} \mathbf{F} = \left(\frac{\mathbf{F} \cdot \mathbf{u_d}}{|\mathbf{u_d}|^2}\right) \mathbf{u_d} \] Then we find the dot product \( \mathbf{F} \cdot \mathbf{d} = (0)(7) + (1)(-9) + (-2)(2) = -9 - 4 = -13 \). Hence, the magnitude of the force in the direction of motion is: \[ |\text{proj}_{\mathbf{d}} \mathbf{F}| = \left|\frac{-13}{\sqrt{134}}\right| \approx 1.1. \] So, \( \mathbf{F} \) has a magnitude of approximately 1.1 in the direction of motion. Since the projection is negative, it indicates that the force is acting opposite to the direction of motion. Therefore, the particle moves along \( -\mathbf{d} \), opposite to the direction of motion. Magnitude of \( \mathbf{F}= \) 1.1 In which direction does the particle move? Along -d, opposite to the direction of motion