Forces of magnitudes 3 and 2 units acting in | vedclass

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(C) The unit vector in the direction of $\vec{a} = 5\hat{i} + 3\hat{j} + 4\hat{k}$ is $\hat{a} = \frac{5\hat{i} + 3\hat{j} + 4\hat{k}}{\sqrt{25 + 9 +

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$\frac{57}{5}\sqrt{2}$ units

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(C) The unit vector in the direction of $\vec{a} = 5\hat{i} + 3\hat{j} + 4\hat{k}$ is $\hat{a} = \frac{5\hat{i} + 3\hat{j} + 4\hat{k}}{\sqrt{25 + 9 + 16}} = \frac{5\hat{i} + 3\hat{j} + 4\hat{k}}{5\sqrt{2}}$.The unit vector in the direction of $\vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ is $\hat{b} = \frac{3\hat{i} + 4\hat{j} - 5\hat{k}}{\sqrt{9 + 16 + 25}} = \frac{3\hat{i} + 4\hat{j} - 5\hat{k}}{5\sqrt{2}}$.The forces are $\vec{F_1} = 3\hat{a} = \frac{3(5\hat{i} + 3\hat{j} + 4\hat{k})}{5\sqrt{2}}$ and $\vec{F_2} = 2\hat{b} = \frac{2(3\hat{i} + 4\hat{j} - 5\hat{k})}{5\sqrt{2}}$.The net force $\vec{F} = \vec{F_1} + \vec{F_2} = \frac{1}{5\sqrt{2}} [(15\hat{i} + 9\hat{j} + 12\hat{k}) + (6\hat{i} + 8\hat{j} - 10\hat{k})] = \frac{21\hat{i} + 17\hat{j} + 2\hat{k}}{5\sqrt{2}}$.The displacement vector $\vec{d} = (3 - 1)\hat{i} + (3 - (-1))\hat{j} + (1 - (-1))\hat{k} = 2\hat{i} + 4\hat{j} + 2\hat{k}$.Work done $W = \vec{F} \cdot \vec{d} = \frac{1}{5\sqrt{2}} (21\hat{i} + 17\hat{j} + 2\hat{k}) \cdot (2\hat{i} + 4\hat{j} + 2\hat{k})$.$W = \frac{1}{5\sqrt{2}} (21 	imes 2 + 17 	imes 4 + 2 	imes 2) = \frac{42 + 68 + 4}{5\sqrt{2}} = \frac{114}{5\sqrt{2}}$.Rationalizing the denominator: $W = \frac{114\sqrt{2}}{5 	imes 2} = \frac{57\sqrt{2}}{5}$ units.

Forces of magnitudes $3$ and $2$ units acting in the directions $5\hat{i} + 3\hat{j} + 4\hat{k}$ and $3\hat{i} + 4\hat{j} - 5\hat{k}$ respectively act on a particle which is displaced from the points $(1, -1, -1)$ to $(3, 3, 1)$. The work done by the forces is equal to

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