To find the magnitude of the force needed to supply 100 newton-meters of torque, we can use the torque equation:

$\text{Torque} = \mathbf{r} \times \mathbf{F}$

Where:

• $\mathbf{r}$ is the position vector of the point where the force is applied (from the origin to the point on the wrench).
• $\mathbf{F}$ is the force vector.
• $\times$ denotes the cross product between the vectors.

Step 1: Define the position vector $\mathbf{r}$

The wrench lies along the positive $y$-axis, and its length is $0.7$ meters. So the position vector is:

$\mathbf{r} = \langle 0, 0.7, 0 \rangle$

Step 2: Define the force vector $\mathbf{F}$

The force is applied in the direction of $\langle 0, -3, -2 \rangle$. To find the actual force, we will assume the force vector $\mathbf{F} = F \langle 0, -3, -2 \rangle$, where $F$ is the magnitude of the force in newtons. This is the direction of the force, but we don't know $F$ yet.

Step 3: Set up the cross product $\mathbf{r} \times \mathbf{F}$

The cross product is computed as:

$\mathbf{r} \times \mathbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0.7 & 0 \\ 0 & -3F & -2F \end{vmatrix}$

Expanding this determinant:

- \hat{j} \left( \begin{vmatrix} 0 & 0 \\ 0 & -2F \end{vmatrix} \right) + \hat{k} \left( \begin{vmatrix} 0 & 0.7 \\ 0 & -3F \end{vmatrix} \right)$$ Solving the determinants: $$\mathbf{r} \times \mathbf{F} = \hat{i} \left( (0.7)(-2F) - (0)(-3F) \right) - \hat{j} \left( (0)(-2F) - (0)(0) \right) + \hat{k} \left( (0)(-3F) - (0)(0.7) \right)$$ $$\mathbf{r} \times \mathbf{F} = \hat{i} \left( -1.4F \right) - \hat{j}(0) + \hat{k}(0)$$ Thus, the torque vector is: $$\mathbf{r} \times \mathbf{F} = \langle -1.4F, 0, 0 \rangle$$ ### Step 4: Set the magnitude of the torque to 100 N·m The magnitude of the torque vector is given by: $$|\mathbf{r} \times \mathbf{F}| = 100$$ The magnitude of the torque vector is: $$| \langle -1.4F, 0, 0 \rangle | = 1.4F$$ So, we set: $$1.4F = 100$$ Solving for $$F$$: $$F = \frac{100}{1.4} = 71.43 \text{ newtons}$$ ### Final Answer: The magnitude of the force needed is **71.43 newtons**. Would you like more details on any step or have any further questions? Here are some follow-up questions to deepen understanding: 1. How do the direction and magnitude of the position vector affect the torque? 2. What would happen if the force were applied at a different angle? 3. How is torque calculated for rotational systems in physics? 4. What role does the angle between the force and the lever arm play in calculating torque? 5. What happens to the torque if the length of the wrench were doubled? **Tip**: When calculating torque, remember that the force vector must be perpendicular to the radius for maximum torque. If the force is not perpendicular, use the cross product to account for the angle.