Step 1: Understanding the Concept:
Torque (\(\vec{\tau}\)) is the rotational equivalent of linear force.
It is defined as the cross product of the position vector (\(\vec{r}\)) and the force vector (\(\vec{F}\)). Step 2: Key Formula or Approach:
The mathematical definition of torque is \( \vec{\tau} = \vec{r} \times \vec{F} \). Step 3: Detailed Explanation:
We are given the vectors: \(\vec{r} = 4\hat{i} - 3\hat{j}\) and \(\vec{F} = 5\hat{i} - 10\hat{j}\).
We need to calculate the cross product: \(\vec{\tau} = (4\hat{i} - 3\hat{j}) \times (5\hat{i} - 10\hat{j})\).
We can compute this using the determinant method or by direct distributive multiplication.
Using the determinant method:
\[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} 4 & -3 & 0 5 & -10 & 0 \end{vmatrix} \]
Expanding along the top row:
\[ \vec{\tau} = \hat{i}((-3)(0) - (0)(-10)) - \hat{j}((4)(0) - (0)(5)) + \hat{k}((4)(-10) - (-3)(5)) \]
The \(\hat{i}\) and \(\hat{j}\) components evaluate to zero.
\[ \vec{\tau} = \hat{k}(-40 - (-15)) \]
\[ \vec{\tau} = \hat{k}(-40 + 15) \]
\[ \vec{\tau} = -25\hat{k} \]
Step 4: Final Answer:
The torque acting on the object is \( -25\hat{k} \).