A force \( \mathbf{F}=[-7,5,9] \) is applied to move a bead on a wire in the direction \( \mathbf{d}=[1,4,-6] \). What is the force in the direction of motion? \( \operatorname{proj}_{\mathbf{d}} \mathbf{F}=[ \) Ex: 1.2

To find the force in the direction of motion, you need to calculate the projection of the force vector \( \mathbf{F} \) onto the direction vector \( \mathbf{d} \). The formula for the projection is: \[ \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \frac{\mathbf{F} \cdot \mathbf{d}}{\mathbf{d} \cdot \mathbf{d}} \mathbf{d} \] First, compute the dot product \( \mathbf{F} \cdot \mathbf{d} \): \[ \mathbf{F} \cdot \mathbf{d} = (-7)(1) + (5)(4) + (9)(-6) = -7 + 20 - 54 = -41 \] Next, compute \( \mathbf{d} \cdot \mathbf{d} \): \[ \mathbf{d} \cdot \mathbf{d} = (1)(1) + (4)(4) + (-6)(-6) = 1 + 16 + 36 = 53 \] Now plug these values into the projection formula: \[ \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \frac{-41}{53} \mathbf{d} \] \[ = \frac{-41}{53} [1, 4, -6] = \left[-\frac{41}{53}, -\frac{164}{53}, \frac{246}{53}\right] \] So, the result of the projection of the force in the direction of motion, expressed in the required format is: \(\operatorname{proj}_{\mathbf{d}} \mathbf{F}=[ -\frac{41}{53}, -\frac{164}{53}, \frac{246}{53}] \)