Let's proceed step-by-step: --- ## **Step 1: Draw the Geometry and Assign Coordinates** Let’s assign coordinates to each point: - \( A = ( 54,) \ mm - B = (, ) \) (This is our reference) - \( C = (15, -21, ) \) mm --- ## **Step 2: Find the Force Vector \( \vec{F} \) along \( AB \)** The force is **2.4 kN** directed *down and to the left along the centerline of AB*. ### **Find the unit vector along \( AB \):** \[ \vec{AB} = \vec{A} - \vec{B} = (-)\mathbf{i} + (54-)\mathbf{j} + (-)\mathbf{k} = 54\mathbf{j} \text{ mm} \] But the picture suggests that the force is acting from A to B, so: \[ \vec{AB} = (-)\mathbf{i} + (-54)\mathbf{j} + (-)\mathbf{k} = -54\mathbf{j} \text{ mm} \] But we need the direction from **A to B**: \[ \vec{AB} = (-)\mathbf{i} + (-54)\mathbf{j} + (-)\mathbf{k} = -54\mathbf{j} \text{ mm} \] ### **Unit vector from A to B:** \[ \vec{u}_{AB} = \frac{\vec{AB}}{|\vec{AB}|} \] \[ |\vec{AB}| = \sqrt{()^2 + (-54)^2 + ()^2} = 54 \text{ mm} \] \[ \vec{u}_{AB} = (, -1, ) \] ### **Force vector (in kN):** \[ \vec{F} = 2.4 \, \text{kN} \cdot (, -1, ) = (, -2.4, ) \, \text{kN} \] --- ## **Step 3: Position Vector from C to B** \[ \vec{r}_{CB} = \vec{B} - \vec{C} = (-15)\mathbf{i} + (+21)\mathbf{j} + (-)\mathbf{k} = (-15, 21, ) \, \text{mm} \] Convert to meters: \[ \vec{r}_{CB} = (-.015, .021, ) \, \text{m} \] --- ## **Step 4: Moment About Point C** \[ \vec{M}_C = \vec{r}_{CB} \times \vec{F} \] \[ \vec{r}_{CB} = (-.015, .021, ) \] \[ \vec{F} = (, -2.4, ) \] Cross product: \[ \vec{M}_C = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -.015 & .021 & \\ & -2.4 & \end{vmatrix} \] Expand: - \( \mathbf{i} \): \( (.021)() - ()(-2.4) = \) - \( \mathbf{j} \): \( -((-.015)() - ()()) = \) - \( \mathbf{k} \): \( (-.015)(-2.4) - (.021)() = .036 \) \[ \vec{M}_C = (, , .036) \, \text{kN}\cdot\text{m} \] --- ## **Step 5: Final Answer** \[ \boxed{ \vec{M}_C = .036\, \mathbf{k}\ \text{kN}\cdot\text{m} } \] - **Direction:** Out of the plane (along \( +z \) axis) - **Magnitude:** \( .036\, \text{kN}\cdot\text{m} \) --- ### **Summary Table** | Step | Calculation | Value | |------|-------------|-------| | \(\vec{r}_{CB}\) | \((B)-(C)\) | \((-.015, .021, )\) m | | \(\vec{F}\) | \(2.4\,\text{kN} \cdot (, -1, )\) | \((, -2.4, )\) kN | | \(\vec{M}_C\) | \(\vec{r}_{CB} \times \vec{F}\) | \((, , .036)\) kN·m | --- If you need any clarification or more details, let me know!