1. **Problem Statement:** We have a pipe assembly fixed in 3D space with points A, B, and C defined as follows: - A at origin (0,0,0) - B is 500 mm in positive x-direction from A - From B, pipe extends 400 mm in positive y-direction, then 300 mm downward (negative z), then 300 mm further in positive y to C A force $\mathbf{F} = 600\mathbf{i} + 800\mathbf{j} - 500\mathbf{k}$ N acts at point C. We need to: (a) Draw a labelled diagram (not done here as per instructions) (b) Find position vector from B to C (c) Find moment of force $\mathbf{F}$ about point B (d) Find magnitude of that moment 2. **Position Vectors:** - Point B coordinates: $\mathbf{r}_B = (500, 0, 0)$ mm - From B to C: - 400 mm in $+y$ direction - 300 mm in $-z$ direction - 300 mm in $+y$ direction So total displacement from B to C: $$\mathbf{r}_{BC} = 0\mathbf{i} + (400 + 300)\mathbf{j} - 300\mathbf{k} = 0\mathbf{i} + 700\mathbf{j} - 300\mathbf{k} \text{ mm}$$ Convert mm to meters for consistency: $$\mathbf{r}_{BC} = 0\mathbf{i} + 0.7\mathbf{j} - 0.3\mathbf{k} \text{ m}$$ 3. **Moment of Force about B:** Moment $\mathbf{M}_B = \mathbf{r}_{BC} \times \mathbf{F}$ Given: $$\mathbf{F} = 600\mathbf{i} + 800\mathbf{j} - 500\mathbf{k} \text{ N}$$ Calculate cross product: $$\mathbf{M}_B = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 0.7 & -0.3 \\ 600 & 800 & -500 \end{vmatrix}$$ Calculate components: - $M_x = (0.7)(-500) - (-0.3)(800) = -350 + 240 = -110$ Nm - $M_y = - \left(0 \cdot (-500) - (-0.3)(600)\right) = - (0 + 180) = -180$ Nm - $M_z = 0 \cdot 800 - 0.7 \cdot 600 = 0 - 420 = -420$ Nm So: $$\mathbf{M}_B = -110\mathbf{i} - 180\mathbf{j} - 420\mathbf{k} \text{ Nm}$$ 4. **Magnitude of Moment:** $$|\mathbf{M}_B| = \sqrt{(-110)^2 + (-180)^2 + (-420)^2} = \sqrt{12100 + 32400 + 176400} = \sqrt{220900} \approx 470.1 \text{ Nm}$$ **Final answers:** - Position vector from B to C: $\boxed{\mathbf{r}_{BC} = 0\mathbf{i} + 0.7\mathbf{j} - 0.3\mathbf{k} \text{ m}}$ - Moment about B: $\boxed{\mathbf{M}_B = -110\mathbf{i} - 180\mathbf{j} - 420\mathbf{k} \text{ Nm}}$ - Magnitude of moment: $\boxed{470.1 \text{ Nm}}$