Please solve all thé question paper in english

The resultant force $\vec{R}$ is the vector sum of $\vec{F_1}$ and $\vec{F_2}$.
Given $\vec{F_1} = (-8\vec{i} + 8\vec{j} - 4\vec{k})$ N and $\vec{F_2} = (-4\vec{i} - 4\vec{j} - 2\vec{k})$ N.

$$\vec{R} = \vec{F_1} + \vec{F_2}$$ $$\vec{R} = (-8\vec{i} + 8\vec{j} - 4\vec{k}) + (-4\vec{i} - 4\vec{j} - 2\vec{k})$$ $$\vec{R} = (-8 - 4)\vec{i} + (8 - 4)\vec{j} + (-4 - 2)\vec{k}$$ $$\vec{R} = -12\vec{i} + 4\vec{j} - 6\vec{k} \text{ N}$$

Since the forces $\vec{F_1}$ and $\vec{F_2}$ are concurrent, their resultant $\vec{R}$ also acts through their point of concurrency.
From the previous part, the point of concurrency P was found using $\lambda_1 = \frac{9}{16}$ with $\vec{r_1}$:

$$\vec{r_P} = \vec{r_1} + \lambda_1 \vec{F_1}$$ $$\vec{r_P} = (5\vec{i} - 6\vec{j} - \vec{k}) + \frac{9}{16}(-8\vec{i} + 8\vec{j} - 4\vec{k})$$ $$\vec{r_P} = (5\vec{i} - 6\vec{j} - \vec{k}) + (-\frac{72}{16}\vec{i} + \frac{72}{16}\vec{j} - \frac{36}{16}\vec{k})$$ $$\vec{r_P} = (5\vec{i} - 6\vec{j} - \vec{k}) + (-\frac{9}{2}\vec{i} + \frac{9}{2}\vec{j} - \frac{9}{4}\vec{k})$$ $$\vec{r_P} = \left(5 - \frac{9}{2}\right)\vec{i} + \left(-6 + \frac{9}{2}\right)\vec{j} + \left(-1 - \frac{9}{4}\right)\vec{k}$$ $$\vec{r_P} = \left(\frac{10 - 9}{2}\right)\vec{i} + \left(\frac{-12 + 9}{2}\right)\vec{j} + \left(\frac{-4 - 9}{4}\right)\vec{k}$$ $$\vec{r_P} = \frac{1}{2}\vec{i} - \frac{3}{2}\vec{j} - \frac{13}{4}\vec{k} \text{ m}$$

The moment of $\vec{R}$ about the origin O is given by $\vec{M_O} = \vec{r_P} \times \vec{R}$.

$$\vec{M_O} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{1}{2} & -\frac{3}{2} & -\frac{13}{4} \\ -12 & 4 & -6 \end{vmatrix}$$ $$\vec{M_O} = \vec{i}\left(\left(-\frac{3}{2}\right)(-6) - \left(-\frac{13}{4}\right)(4)\right) - \vec{j}\left(\left(\frac{1}{2}\right)(-6) - \left(-\frac{13}{4}\right)(-12)\right) + \vec{k}\left(\left(\frac{1}{2}\right)(4) - \left(-\frac{3}{2}\right)(-12)\right)$$ $$\vec{M_O} = \vec{i}\left(9 - (-13)\right) - \vec{j}\left(-3 - (39)\right) + \vec{k}\left(2 - (18)\right)$$ $$\vec{M_O} = \vec{i}(9 + 13) - \vec{j}(-3 - 39) + \vec{k}(2 - 18)$$ $$\vec{M_O} = 22\vec{i} + 42\vec{j} - 16\vec{k} \text{ N}\cdot\text{m}$$

The resultant force is $\boxed{\vec{R} = -12\vec{i} + 4\vec{j} - 6\vec{k} \text{ N}}$.
The moment of $\vec{R}$ about the origin O is $\boxed{\vec{M_O} = 22\vec{i} + 42\vec{j} - 16\vec{k} \text{ N}\cdot\text{m}}$.