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

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Let the force be $$\mathbf{F} = [-7, 5, 9]$$ and the direction vector be $$\mathbf{d} = [1, 4, -6]$$.

1. Compute the dot product $$\mathbf{F} \cdot \mathbf{d}$$:
   $$
   \mathbf{F} \cdot \mathbf{d} = (-7)(1) + (5)(4) + (9)(-6) = -7 + 20 - 54 = -41.
   $$

2. Compute the magnitude squared of $$\mathbf{d}$$:
   $$
   \|\mathbf{d}\|^2 = 1^2 + 4^2 + (-6)^2 = 1 + 16 + 36 = 53.
   $$

3. The projection of $$\mathbf{F}$$ in the direction of $$\mathbf{d}$$ is then given by:
   $$
   \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \frac{\mathbf{F} \cdot \mathbf{d}}{\|\mathbf{d}\|^2} \mathbf{d} = \frac{-41}{53} \, [1, 4, -6].
   $$

4. Multiply the scalar component with the vector $$\mathbf{d}$$:
   $$
   \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \left[ -\frac{41}{53} \cdot 1, \; -\frac{41}{53} \cdot 4, \; -\frac{41}{53} \cdot (-6) \right].
   $$
   $$
   \operatorname{proj}_{\mathbf{d}} \mathbf{F} = \left[ -\frac{41}{53}, \; -\frac{164}{53}, \; \frac{246}{53} \right].
   $$

The force in the direction of motion is:
$$
\operatorname{proj}_{\mathbf{d}} \mathbf{F} = \left[ -\frac{41}{53}, -\frac{164}{53}, \frac{246}{53} \right].
$$
The force in the direction of motion is $$\left[ -\frac{41}{53}, -\frac{164}{53}, \frac{246}{53} \right]$$.

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

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