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To solve this problem, we need to calculate the torque produced by the force applied at an angle on the wrench.

### Given Information
- Force vector, \( \vec{F} = (2, 10, 11) \) Newtons.
- Length of the wrench, \( r = 0.45 \) meters.
- Angle between the force vector and the wrench, \( \theta = 35^\circ \).

### Torque Calculation
Torque (\( \tau \)) is calculated by the formula:
\[
\tau = r \cdot F \cdot \sin(\theta)
\]
where:
- \( r \) is the length of the wrench,
- \( F \) is the magnitude of the force vector,
- \( \theta \) is the angle between the force vector and the wrench.

#### Step 1: Calculate the Magnitude of the Force Vector
To find \( F \), we calculate the magnitude of the force vector \( \vec{F} = (2, 10, 11) \):
\[
F = \sqrt{2^2 + 10^2 + 11^2}
\]
\[
F = \sqrt{4 + 100 + 121} = \sqrt{225} = 15 \text{ N}
\]

#### Step 2: Calculate the Torque
Using the torque formula:
\[
\tau = 0.45 \cdot 15 \cdot \sin(35^\circ)
\]

First, find \( \sin(35^\circ) \):
\[
\sin(35^\circ) \approx 0.5736
\]

Then, plug in the values:
\[
\tau = 0.45 \cdot 15 \cdot 0.5736
\]
\[
\tau \approx 3.87 \text{ Nm}
\]

### Answer
The magnitude of the torque is approximately **3.87 Nm**.

Would you like further explanation on any steps?

Here are some related questions for further exploration:
1. How would the torque change if the angle \( \theta \) was increased to 90°?
2. What would be the result if the length of the wrench was doubled?
3. How does changing the direction of the force vector affect the torque?
4. Can torque be negative, and if so, what does that indicate?
5. How does torque relate to rotational equilibrium?

**Tip:** Remember that torque is maximum when the angle between the force and the lever arm is 90°, as \( \sin(90^\circ) = 1 \).

Suppose you apply a force (in Newtons) described by F = (2, 10, 11) to a 0.45 meter long wrench at an angle of θ = 35°. Find the magnitude of the torque that you apply. Round your answer to the nearest hundredth.

Learn how to calculate the torque applied by a force vector F = (2, 10, 11) on a 0.45 meter long wrench at a 35° angle. This problem explores vector magnitude, trigonometry, and the torque formula, providing a step-by-step solution. Ideal for Grades 10-12.

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