1. **Problem statement:** Calculate the moment about the origin $O$ of the force $\mathbf{F} = -2\mathbf{i} - 3\mathbf{j} + 5\mathbf{k}$ acting at point $A$ with given position vectors. 2. **Formula:** The moment $\mathbf{M}_O$ about the origin is given by the cross product: $$\mathbf{M}_O = \mathbf{r} \times \mathbf{F}$$ where $\mathbf{r}$ is the position vector of point $A$. 3. **Cross product rule:** For vectors $\mathbf{a} = a_x\mathbf{i} + a_y\mathbf{j} + a_z\mathbf{k}$ and $\mathbf{b} = b_x\mathbf{i} + b_y\mathbf{j} + b_z\mathbf{k}$, $$\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y)\mathbf{i} - (a_x b_z - a_z b_x)\mathbf{j} + (a_x b_y - a_y b_x)\mathbf{k}$$ 4. **Calculate moments for each case:** (a) $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k}$ $$\mathbf{M}_O = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 1 \\ -2 & -3 & 5 \end{vmatrix} = ((1)(5) - (1)(-3))\mathbf{i} - ((1)(5) - (1)(-2))\mathbf{j} + ((1)(-3) - (1)(-2))\mathbf{k}$$ $$= (5 + 3)\mathbf{i} - (5 + 2)\mathbf{j} + (-3 + 2)\mathbf{k} = 8\mathbf{i} - 7\mathbf{j} - 1\mathbf{k}$$ (b) $\mathbf{r} = 4\mathbf{i} + 6\mathbf{j} - 10\mathbf{k}$ $$\mathbf{M}_O = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 6 & -10 \\ -2 & -3 & 5 \end{vmatrix} = (6 \times 5 - (-10) \times (-3))\mathbf{i} - (4 \times 5 - (-10) \times (-2))\mathbf{j} + (4 \times (-3) - 6 \times (-2))\mathbf{k}$$ $$= (30 - 30)\mathbf{i} - (20 - 20)\mathbf{j} + (-12 + 12)\mathbf{k} = 0\mathbf{i} - 0\mathbf{j} + 0\mathbf{k} = \mathbf{0}$$ (c) $\mathbf{r} = 4\mathbf{i} + 3\mathbf{j} - 5\mathbf{k}$ $$\mathbf{M}_O = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 3 & -5 \\ -2 & -3 & 5 \end{vmatrix} = (3 \times 5 - (-5) \times (-3))\mathbf{i} - (4 \times 5 - (-5) \times (-2))\mathbf{j} + (4 \times (-3) - 3 \times (-2))\mathbf{k}$$ $$= (15 - 15)\mathbf{i} - (20 - 10)\mathbf{j} + (-12 + 6)\mathbf{k} = 0\mathbf{i} - 10\mathbf{j} - 6\mathbf{k}$$ 5. **Final answers:** - (a) $\mathbf{M}_O = 8\mathbf{i} - 7\mathbf{j} - \mathbf{k}$ - (b) $\mathbf{M}_O = \mathbf{0}$ - (c) $\mathbf{M}_O = -10\mathbf{j} - 6\mathbf{k}$