1. **State the problem:** Calculate the work done $W$ by force $\mathbf{F}_1 = 5\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$ acting through displacement $\mathbf{S} = 3\mathbf{i} + 4\mathbf{j} - \mathbf{k}$, and find the angle $\theta$ between these two vectors. 2. **Formula for work done:** $$W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}| |\mathbf{d}| \cos \theta$$ The dot product $\mathbf{F} \cdot \mathbf{d}$ equals the scalar work done. 3. **Calculate the dot product:** $$W = (5)(3) + (-2)(4) + (3)(-1) = 15 - 8 - 3 = 4$$ So, the work done is $4$ (units depend on context). 4. **Calculate magnitudes:** $$|\mathbf{F}_1| = \sqrt{5^2 + (-2)^2 + 3^2} = \sqrt{25 + 4 + 9} = \sqrt{38}$$ $$|\mathbf{S}| = \sqrt{3^2 + 4^2 + (-1)^2} = \sqrt{9 + 16 + 1} = \sqrt{26}$$ 5. **Find the angle $\theta$ between vectors:** Using the dot product formula: $$\mathbf{F}_1 \cdot \mathbf{S} = |\mathbf{F}_1| |\mathbf{S}| \cos \theta$$ Rearranged: $$\cos \theta = \frac{\mathbf{F}_1 \cdot \mathbf{S}}{|\mathbf{F}_1| |\mathbf{S}|} = \frac{4}{\sqrt{38} \times \sqrt{26}} = \frac{4}{\sqrt{988}}$$ 6. **Calculate $\theta$:** $$\theta = \cos^{-1} \left( \frac{4}{\sqrt{988}} \right) \approx \cos^{-1}(0.1274) \approx 82.68^\circ$$ **Final answers:** - Work done $W = 4$ - Angle between vectors $\theta \approx 82.68^\circ$