A particle acted on by constant forces 4\widehat i + \widehat j - 3\widehat k and 3\widehat i + \widehat j - \widehat k is displaced from the point \widehat i + 2\widehat j + 3\widehat k to the point \,5\widehat i + 4\widehat j + \widehat k. The total work done by the forces is :

Vector Algebra PYQ — JEE Main 2004 · Shift 0 · Q91 with Solution

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Correct answer: D

The two constant forces are $\vec F_1 = 4\hat i + \hat j - 3\hat k$ and $\vec F_2 = 3\hat i + \hat j - \hat k.$

Since both forces act simultaneously, the resultant force is $\vec F = \vec F_1 + \vec F_2.$

So, $\vec F = (4+3)\hat i + (1+1)\hat j + (-3-1)\hat k$ $\vec F = 7\hat i + 2\hat j - 4\hat k.$

Initial position: $\vec r_1 = \hat i + 2\hat j + 3\hat k$

Final position: $\vec r_2 = 5\hat i + 4\hat j + \hat k$

Therefore displacement is $\vec s = \vec r_2 - \vec r_1$ $\vec s = (5-1)\hat i + (4-2)\hat j + (1-3)\hat k$ $\vec s = 4\hat i + 2\hat j - 2\hat k.$

Total work done is the dot product of resultant force and displacement: $W = \vec F \cdot \vec s$

So, $W = (7\hat i + 2\hat j - 4\hat k) \cdot (4\hat i + 2\hat j - 2\hat k)$

$W = 7\cdot 4 + 2\cdot 2 + (-4)\cdot (-2)$ $W = 28 + 4 + 8$ $W = 40.$

Thus, the total work done is $\boxed{40\text{ units}}.$

So the correct option is D.