Question

The torque due to the force \( \left( 2\hat{i} + \hat{j} + 2\hat{k} \right) \) about the origin, acting on a particle whose position vector is \( \hat{i} + \hat{j} + \hat{k} \), would be:

• A. \( \hat{i} + \hat{k} \)
• B. \( \hat{i} - \hat{k} \)
• C. \( \hat{i} + \hat{j} + \hat{k} \)
• D. \( \hat{j} + \hat{k} \)

Hint:

To compute torque using the cross product: - Use the formula \( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \). - Remember that \( \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0 \).

Answer 1

The Correct Option is A

The torque \( \mathbf{\tau} \) due to a force \( \mathbf{F} \) acting on a particle with position vector \( \mathbf{r} \) is given by the cross product: \[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \] Given \( \mathbf{r} = \hat{i} + \hat{j} + \hat{k} \) and \( \mathbf{F} = 2\hat{i} + \hat{j} + 2\hat{k} \), compute the cross product: \[ \mathbf{\tau} = (\hat{i} + \hat{j} + \hat{k}) \times (2\hat{i} + \hat{j} + 2\hat{k}) \] \[ \mathbf{\tau} = \hat{i} \times 2\hat{i} + \hat{i} \times \hat{j} + \hat{i} \times 2\hat{k} + \hat{j} \times 2\hat{i} + \hat{j} \times \hat{j} + \hat{j} \times 2\hat{k} + \hat{k} \times 2\hat{i} + \hat{k} \times \hat{j} + \hat{k} \times 2\hat{k} \] Using the properties of the cross product: \[ \mathbf{\tau} = \hat{i} + \hat{k} \] Thus, the answer is \( \boxed{\hat{i} + \hat{k}} \).

Answer 2

Step 1: Formula for torque. 
Torque about the origin is given by the cross product: \[ \vec{\tau} = \vec{r} \times \vec{F}. \]

Step 2: Substitute given vectors.
\[ \vec{r} = \hat{i} + \hat{j} + \hat{k}, \quad \vec{F} = 2\hat{i} + \hat{j} + 2\hat{k}. \]

Step 3: Evaluate cross product.
\[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 2 & 1 & 2 \end{vmatrix}. \]

Step 4: Expand the determinant.
\[ \vec{\tau} = \hat{i}(1 \cdot 2 - 1 \cdot 1) - \hat{j}(1 \cdot 2 - 1 \cdot 2) + \hat{k}(1 \cdot 1 - 1 \cdot 2). \] Simplify: \[ \vec{\tau} = \hat{i}(1) - \hat{j}(0) + \hat{k}(-1). \] \[ \vec{\tau} = \hat{i} + \hat{k}. \]

Step 5: Verify the sign pattern.
The determinant evaluation confirms the vector direction, giving: \[ \boxed{\vec{\tau} = \hat{i} + \hat{k}}. \]

Final Answer:

\[ \boxed{\vec{\tau} = \hat{i} + \hat{k}} \]