Question: Determine the magnitude of the component of force acting perpendicular to line $AO$.
Express your answer to three significant figures and include the appropriate units.
(F_{AO})_\perp = Value N
1. **Problem Statement:**
We need to find the magnitude of the component of a force acting perpendicular to the line $AO$.
2. **Understanding the problem:**
The force $\mathbf{F}$ acts along a cable from point $A$ toward point $D$. We want the component of this force that is perpendicular to the line segment $AO$.
3. **Formula and approach:**
The force vector $\mathbf{F}$ can be decomposed into two components relative to $AO$:
- Parallel component: $\mathbf{F}_{\parallel}$ along $AO$
- Perpendicular component: $\mathbf{F}_{\perp}$ perpendicular to $AO$
The magnitude of the perpendicular component is given by:
$$
F_{\perp} = |\mathbf{F}| \sin \theta
$$
where $\theta$ is the angle between $\mathbf{F}$ and the line $AO$.
Alternatively, if $\mathbf{F}$ and the unit vector along $AO$, $\hat{u}_{AO}$, are known, then:
$$
F_{\perp} = \sqrt{|\mathbf{F}|^2 - (\mathbf{F} \cdot \hat{u}_{AO})^2}
$$
4. **Steps to solve:**
- Find the vector $\mathbf{AO}$ from coordinates of points $A$ and $O$.
- Calculate the unit vector $\hat{u}_{AO} = \frac{\mathbf{AO}}{|\mathbf{AO}|}$.
- Find the force vector $\mathbf{F}$ along $AD$.
- Compute the dot product $\mathbf{F} \cdot \hat{u}_{AO}$.
- Calculate $F_{\perp} = \sqrt{|\mathbf{F}|^2 - (\mathbf{F} \cdot \hat{u}_{AO})^2}$.
5. **Explanation:**
The perpendicular component is the part of the force that does not act along $AO$. It is found by subtracting the squared parallel component from the total squared magnitude of the force and then taking the square root.
6. **Final answer:**
Express $F_{\perp}$ to three significant figures and include units (N).
*Note: Specific coordinates and force magnitude are needed to compute a numeric answer, which are not provided in the prompt.*