Let's break the problem into **two steps**: 1. **Find the position vector \( \vec{r} \) from the origin to point P.** 2. **Calculate the torque \( \vec{\tau} = \vec{r} \times \vec{F} \) about the origin.** --- ### **Step 1: Find the Position Vector \(\vec{r}\) to Point P** Pipe lengths: - \( L_1 = 1.45\, \text{m} \) (along the x-axis) - \( L_2 = .650\, \text{m} \) (along the y-axis) - \( L_3 = 2.70\, \text{m} \) (along the z-axis) So, \[ \vec{r} = L_1\hat{i} + L_2\hat{j} + L_3\hat{k} = (1.45 \hat{i} + .650 \hat{j} + 2.70 \hat{k})~\text{m} \] > **Explanation:** > The position vector is just the sum of the displacements along each axis, so we add each segment in its respective direction. --- ### **Step 2: Calculate the Torque \(\vec{\tau} = \vec{r} \times \vec{F}\)** Given force at P: \[ \vec{F} = (-23.5 \hat{i} + 47.5 \hat{j} + 28. \hat{k})~\text{N} \] The cross product \(\vec{\tau} = \vec{r} \times \vec{F}\): \[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1.45 & .650 & 2.70 \\ -23.5 & 47.5 & 28. \\ \end{vmatrix} \] Expanding the determinant: - \( \hat{i}: (.650 \cdot 28. - 2.70 \cdot 47.5) \) - \( \hat{j}: -(1.45 \cdot 28. - 2.70 \cdot -23.5) \) - \( \hat{k}: (1.45 \cdot 47.5 - .650 \cdot -23.5) \) Compute each component: 1. **\( \hat{i} \):** \[ .650 \cdot 28. = 18.2 \] \[ 2.70 \cdot 47.5 = 128.25 \] \[ 18.2 - 128.25 = -110.05 \] 2. **\( \hat{j} \):** \[ 1.45 \cdot 28. = 40.6 \] \[ 2.70 \cdot (-23.5) = -63.45 \] \[ 40.6 - (-63.45) = 40.6 + 63.45 = 104.05; \text{but overall negative: } -104.05 \] 3. **\( \hat{k} \):** \[ 1.45 \cdot 47.5 = 68.875 \] \[ .650 \cdot (-23.5) = -15.275 \] \[ 68.875 - (-15.275) = 68.875 + 15.275 = 84.15 \] So, \[ \boxed{ \vec{\tau} = -110\hat{i} - 104\hat{j} + 84.2\hat{k}~\text{N}\cdot\text{m} } \] (Rounded to three significant figures.) > **Explanation:** > The torque is found using the cross product of position and force vectors. Each component is calculated using the determinant expansion. --- ## **Final Answer** \[ \boxed{ \vec{\tau} = -110\hat{i} - 104\hat{j} + 84.2\hat{k}~\text{N}\cdot\text{m} } \]