A force \( \mathbf{F}=[-5,7,0] \) is applied to move a bead on a wire in the direction \( \mathbf{d}=[8,5,-7] \). What is the force in the direction of motion? \( \operatorname{proj}_{d} \mathbf{F}=[ \) Ex. 1.2\( \left.], \square\right] \)

To find the force in the direction of motion, we need to calculate the projection of the force \(\mathbf{F}\) onto the direction vector \(\mathbf{d}\). The formula for the projection of \(\mathbf{F}\) onto \(\mathbf{d}\) is given by: \[ \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \frac{\mathbf{F} \cdot \mathbf{d}}{\mathbf{d} \cdot \mathbf{d}} \mathbf{d} \] Calculating the dot products, we have: \[ \mathbf{F} \cdot \mathbf{d} = [-5, 7, 0] \cdot [8, 5, -7] = (-5 \times 8) + (7 \times 5) + (0 \times -7) = -40 + 35 + 0 = -5 \] Next, we calculate \(\mathbf{d} \cdot \mathbf{d}\): \[ \mathbf{d} \cdot \mathbf{d} = [8, 5, -7] \cdot [8, 5, -7] = 8^2 + 5^2 + (-7)^2 = 64 + 25 + 49 = 138 \] Now, we can find the projection: \[ \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \frac{-5}{138} [8, 5, -7] = \left[ \frac{-40}{138}, \frac{-25}{138}, \frac{35}{138} \right] \] Thus, the force in the direction of motion is: \[ \operatorname{proj}_{d} \mathbf{F} = \left[ \frac{-40}{138}, \frac{-25}{138}, \frac{35}{138} \right] \] Final answer in the required format: \(\operatorname{proj}_{d} \mathbf{F}=[\frac{-20}{69},\frac{-25}{138},\frac{35}{138}]\)