The work done by the forces $P=2\hat{i}-3\hat{j}+\hat{k}$ and $Q=\hat{i}+5\hat{j}-3\hat{k}$ to displace the particle from point A to point B whose position vectors are $-2\hat{i}+5\hat{j}+7\hat{k}$ and $3\hat{i}+7\hat{j}+2\hat{k}$ respectively, is :

Question

Accepted Answer

To determine the work done by the forces $ P $ and $ Q $ in displacing the particle from point A to point B, we need to calculate the dot product of the resultant force vector with the displacement vector.

1. **Calculate the displacement vector**:
   - The position vector of point A is $ \vec{r}_A = -2\hat{i} + 5\hat{j} + 7\hat{k} $.
   - The position vector of point B is $ \vec{r}_B = 3\hat{i} + 7\hat{j} + 2\hat{k} $.
   - The displacement vector $ \vec{d} $ is given by:
     $$
     \vec{d} = \vec{r}_B - \vec{r}_A = (3 - (-2))\hat{i} + (7 - 5)\hat{j} + (2 - 7)\hat{k} = 5\hat{i} + 2\hat{j} - 5\hat{k}
     $$

2. **Calculate the resultant force vector**:
   - The force vector $ P $ is $ 2\hat{i} - 3\hat{j} + \hat{k} $.
   - The force vector $ Q $ is $ \hat{i} + 5\hat{j} - 3\hat{k} $.
   - The resultant force vector $ \vec{F} $ is:
     $$
     \vec{F} = P + Q = (2 + 1)\hat{i} + (-3 + 5)\hat{j} + (1 - 3)\hat{k} = 3\hat{i} + 2\hat{j} - 2\hat{k}
     $$

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

Therefore, the work done by the forces is 29 units, which corresponds to Option 3.

**Explanation of Incorrect Options**:
- **Option 1: $\sqrt{17}$ units**: This option is incorrect because the calculation of the dot product does not result in a square root value. The dot product directly gives a scalar value, which is 29 in this case.
- **Option 2: 31 units**: This option is incorrect because the correct calculation of the dot product yields 29, not 31. The arithmetic operations involved in the dot product do not sum to 31.
- **Option 4: $\sqrt{54}$ units**: Similar to Option 1, this option is incorrect because the result of the dot product is a scalar value, not a square root. The calculation does not involve any square root operations.

Thus, the correct answer is indeed Option 3, as the work done is 29 units.