Example

Question:

Find the torque of a force \( \mathbf{F} = 7\mathbf{\hat{i}} + 3\mathbf{\hat{j}} - 5\mathbf{\hat{k}} \) about the origin.
The force acts on a particle whose position vector is \( \mathbf{r} = \mathbf{\hat{i}} - \mathbf{\hat{j}} + \mathbf{\hat{k}} \).

Solution:

We use the cross product \( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \):

\[ \mathbf{\tau} = \begin{vmatrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ 1 & -1 & 1 \\ 7 & 3 & -5 \\ \end{vmatrix} \]
Calculate each component:
\[ \mathbf{\tau} = ( -1 \times -5 - 1 \times 3 ) \mathbf{\hat{i}} - ( 1 \times -5 - 1 \times 7 ) \mathbf{\hat{j}} + ( 1 \times 3 - ( -1 \times 7 ) ) \mathbf{\hat{k}} \] \[ = (5 - 3)\,\mathbf{\hat{i}} - ( -5 - 7 )\,\mathbf{\hat{j}} + ( 3 + 7 )\,\mathbf{\hat{k}} \] \[ = 2\mathbf{\hat{i}} + 12\mathbf{\hat{j}} + 10\mathbf{\hat{k}} \]
Final Answer:
\( \mathbf{\tau} = 2\mathbf{\hat{i}} + 12\mathbf{\hat{j}} + 10\mathbf{\hat{k}} \)