Question

If force $\vec{F} = -3\hat{i} + \hat{j} + 5\hat{k}$ acts along $\vec{r} = 7\hat{i} + 3\hat{j} + \hat{k}$ then the torque acting at that point is

• A. $(14\hat{i} - 38\hat{j} + 16\hat{k})$
• B. $(-14\hat{i} + 34\hat{j} - 16\hat{k})$
• C. $(21\hat{i} + 4\hat{j} + 4\hat{k})$
• D. $(4\hat{i} + 4\hat{j} + 6\hat{k})$

Hint:

For torque: \[ \vec{\tau}=\vec{r}\times \vec{F} \] Be careful with the minus sign in the \(\hat{j}\)-component while expanding the determinant.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Torque is the vector product of the position vector and the force vector.
Step 2: Key Formula or Approach:
$\vec{\tau} = \vec{r} \times \vec{F}$.
Step 3: Detailed Explanation:
$\vec{r} = (7, 3, 1)$ and $\vec{F} = (-3, 1, 5)$.
$\vec{\tau} = \hat{i}(3 \times 5 - 1 \times 1) - \hat{j}(7 \times 5 - 1 \times -3) + \hat{k}(7 \times 1 - 3 \times -3)$
$\vec{\tau} = \hat{i}(14) - \hat{j}(38) + \hat{k}(16) = 14\hat{i} - 38\hat{j} + 16\hat{k}$.
Step 4: Final Answer:
The torque is $(14\hat{i} - 38\hat{j} + 16\hat{k})$.