The work done by the force $3\hat{i}+2\hat{j}-2\hat{k}$ to displace a particle from point $A(-2\hat{i}+5\hat{j}+7\hat{k})$ to point $B(3\hat{i}+7\hat{j}+2\hat{k})$ is:

Q: The work done by the force $3\hat{i}+2\hat{j}-2\hat{k}$ to displace a particle from point $A(-2\hat{i}+5\hat{j}+7\hat{k})$ to point $B(3\hat{i}+7\hat{j}+2\hat{k})$ is:

To determine the work done by the force vector $ \mathbf{F} = 3\hat{i} + 2\hat{j} - 2\hat{k} $ in displacing a particle from point $ A(-2\hat{i} + 5\hat{j} + 7\hat{k}) $ to point $ B(3\hat{i} + 7\hat{j} + 2\hat{k}) $, we need to calculate the dot product of the force vector with the displacement vector. 1. **Calculate the Displacement Vector:** The displacement vector $ \mathbf{d} $ from point $ A $ to point $ B $ is given by: \[ \mathbf{d} = (3\hat{i} + 7\hat{j} + 2\hat{k}) - (-2\hat{i} + 5\hat{j} + 7\hat{k}) \] \[ \mathbf{d} = (3 + 2)\hat{i} + (7 - 5)\hat{j} + (2 - 7)\hat{k} \] \[ \mathbf{d} = 5\hat{i} + 2\hat{j} - 5\hat{k} \] 2. **Calculate the Work Done:** The work done $ W $ is the dot product of the force vector $ \mathbf{F} $ and the displacement vector $ \mathbf{d} $: \[ W = \mathbf{F} \cdot \mathbf{d} = (3\hat{i} + 2\hat{j} - 2\hat{k}) \cdot (5\hat{i} + 2\hat{j} - 5\hat{k}) \] \[ W = (3 \times 5) + (2 \times 2) + (-2 \times -5) \] \[ W = 15 + 4 + 10 \] \[ W = 29 \text{ Units} \] Therefore, the work done by the force is 29 units, which corresponds to Option 2. 3. **Explanation of Incorrect Options:** - **Option 1 (31 Units):** This option is incorrect because the calculation of the dot product does not yield 31. The correct calculation results in 29 units. - **Option 3 (9 Units):** This option is incorrect as it significantly underestimates the work done. The dot product calculation clearly shows that the work done is 29 units, not 9. - **Option 4 (30 Units):** This option is incorrect because the calculated work done is 29 units. The discrepancy arises from a potential miscalculation or misunderstanding of the dot product operation. In conclusion, the correct answer is Option 2, which is supported by the accurate calculation of the dot product between the force and displacement vectors.

Question

Accepted Answer

To determine the work done by the force vector $ \mathbf{F} = 3\hat{i} + 2\hat{j} - 2\hat{k} $ in displacing a particle from point $ A(-2\hat{i} + 5\hat{j} + 7\hat{k}) $ to point $ B(3\hat{i} + 7\hat{j} + 2\hat{k}) $, we need to calculate the dot product of the force vector with the displacement vector.

1. **Calculate the Displacement Vector:**

The displacement vector $ \mathbf{d} $ from point $ A $ to point $ B $ is given by:
   $$
   \mathbf{d} = (3\hat{i} + 7\hat{j} + 2\hat{k}) - (-2\hat{i} + 5\hat{j} + 7\hat{k})
   $$
   $$
   \mathbf{d} = (3 + 2)\hat{i} + (7 - 5)\hat{j} + (2 - 7)\hat{k}
   $$
   $$
   \mathbf{d} = 5\hat{i} + 2\hat{j} - 5\hat{k}
   $$

2. **Calculate the Work Done:**

The work done $ W $ is the dot product of the force vector $ \mathbf{F} $ and the displacement vector $ \mathbf{d} $:
   $$
   W = \mathbf{F} \cdot \mathbf{d} = (3\hat{i} + 2\hat{j} - 2\hat{k}) \cdot (5\hat{i} + 2\hat{j} - 5\hat{k})
   $$
   $$
   W = (3 \times 5) + (2 \times 2) + (-2 \times -5)
   $$
   $$
   W = 15 + 4 + 10
   $$
   $$
   W = 29 \text{ Units}
   $$

Therefore, the work done by the force is 29 units, which corresponds to Option 2.

3. **Explanation of Incorrect Options:**

- **Option 1 (31 Units):** This option is incorrect because the calculation of the dot product does not yield 31. The correct calculation results in 29 units.
   
   - **Option 3 (9 Units):** This option is incorrect as it significantly underestimates the work done. The dot product calculation clearly shows that the work done is 29 units, not 9.
   
   - **Option 4 (30 Units):** This option is incorrect because the calculated work done is 29 units. The discrepancy arises from a potential miscalculation or misunderstanding of the dot product operation.

In conclusion, the correct answer is Option 2, which is supported by the accurate calculation of the dot product between the force and displacement vectors.