A force \( \mathbf{F}=[0,8,4] \) is applied to move a bead on a wire in the direction \( \mathbf{d}=[3,9,-9] \). 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, we need to calculate the projection of the force \( \mathbf{F} \) onto the direction \( \mathbf{d} \). The formula for the projection of a vector \( \mathbf{a} \) onto a vector \( \mathbf{b} \) is given by: \[ \operatorname{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \] First, we compute the dot products: 1. Compute \( \mathbf{F} \cdot \mathbf{d} \): \[ \mathbf{F} \cdot \mathbf{d} = [0, 8, 4] \cdot [3, 9, -9] = 0 \cdot 3 + 8 \cdot 9 + 4 \cdot (-9) = 0 + 72 - 36 = 36 \] 2. Compute \( \mathbf{d} \cdot \mathbf{d} \): \[ \mathbf{d} \cdot \mathbf{d} = [3, 9, -9] \cdot [3, 9, -9] = 3^2 + 9^2 + (-9)^2 = 9 + 81 + 81 = 171 \] Now we can find the projection: \[ \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \frac{36}{171} \mathbf{d} \] This can be simplified by calculating \( \frac{36}{171} \): \[ \frac{36}{171} = \frac{12}{57} = \frac{4}{19} \] Thus, the projection is: \[ \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \frac{4}{19} [3, 9, -9] = \left[\frac{12}{19}, \frac{36}{19}, -\frac{36}{19}\right] \] So, the answer is: \[ \operatorname{proj}_{\mathbf{d}} \mathbf{F}=\left[\frac{12}{19}, \frac{36}{19}, -\frac{36}{19}\right] \]