https://file.aiquickdraw.com/mathbot/file/172611162736458spa6q4.jpg

The problem in the image is asking to find the torque \(\vec{\tau}\) about the point \(O\), given:

- The position vector \(\vec{r} = -\hat{i} + \hat{j} - \hat{k} = (-1, 1, -1)\),
- The force vector \(\vec{F} = (17, 0, 0)\), applied at point \(P\).

The torque \(\vec{\tau}\) is calculated using the cross product formula:

\[
\vec{\tau} = \vec{r} \times \vec{F}
\]

Let’s compute the cross product of \(\vec{r}\) and \(\vec{F}\).

\[
\vec{r} \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -1 & 1 & -1 \\ 17 & 0 & 0 \end{vmatrix}
\]

To calculate the determinant:

\[
\vec{\tau} = \hat{i} \left( 1 \cdot 0 - (-1) \cdot 0 \right) - \hat{j} \left( -1 \cdot 0 - (-1) \cdot 17 \right) + \hat{k} \left( -1 \cdot 0 - 1 \cdot 17 \right)
\]

Simplifying the components:

\[
\vec{\tau} = \hat{i} (0) - \hat{j} (17) + \hat{k} (-17)
\]

Thus, the torque vector is:

\[
\vec{\tau} = -17 \hat{j} - 17 \hat{k}
\]

Or in component form:

\[
\vec{\tau} = (0, -17, -17)
\]

So, the torque about point \(O\) is \(\vec{\tau} = (0, -17, -17)\).

Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this topic:
1. What is the physical significance of torque?
2. How does the direction of the force vector affect the resulting torque?
3. What is the difference between torque and force?
4. Can torque be zero even if a force is applied? Under what conditions?
5. How do you compute the magnitude of the torque vector?

**Tip**: The direction of the torque is perpendicular to the plane formed by the position and force vectors.

Let r = OP = -i + j - k. A force F = (17, 0, 0) is applied at P. Find the torque τ about O that is produced.

Learn how to calculate the torque about point O using the cross product of the position vector and the force vector. This problem involves vector calculus and is suitable for undergraduate students studying physics or engineering.

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