# Explanation of Problem 3 Let's break down the problem step by step! --- ## **Given:** - **Lengths:** \( L_1, L_2, L_3 \) - **Angles:** \( \theta_1, \theta_2 \) - **Task:** Find the position vector of tip \( R \) with respect to the origin \( O \). - **Form:** \( \vec{r}_{R/O} = (\text{_____})\hat{n}_x + (\text{_____})\hat{n}_y \) --- ## **Step-by-Step Solution** ### **1. Analyzing the Mechanism** - **First Link:** From \( O \) to \( P \) of length \( L_1 \) along \( \hat{n}_x \). - **Second Link:** From \( P \) to \( Q \) of length \( L_2 \) at angle \( \theta_1 \) from \( \hat{n}_x \). - **Third Link:** From \( Q \) to \( R \) of length \( L_3 \) at angle \( \theta_1 + \theta_2 \) from \( \hat{n}_x \). ### **2. Writing Each Vector** #### **From \( O \) to \( P \):** \[ \vec{r}_{P/O} = L_1 \hat{n}_x \] #### **From \( P \) to \( Q \):** \[ \vec{r}_{Q/P} = L_2 (\cos\theta_1 \hat{n}_x + \sin\theta_1 \hat{n}_y) \] #### **From \( Q \) to \( R \):** \[ \vec{r}_{R/Q} = L_3 \left[\cos(\theta_1 + \theta_2) \hat{n}_x + \sin(\theta_1 + \theta_2) \hat{n}_y \right] \] ### **3. Total Position Vector \( \vec{r}_{R/O} \):** Add up all vectors: \[ \vec{r}_{R/O} = \vec{r}_{P/O} + \vec{r}_{Q/P} + \vec{r}_{R/Q} \] Substitute the expressions: \[ \begin{align*} \vec{r}_{R/O} = &\ L_1 \hat{n}_x \\ &+ L_2 (\cos\theta_1 \hat{n}_x + \sin\theta_1 \hat{n}_y) \\ &+ L_3 [\cos(\theta_1 + \theta_2) \hat{n}_x + \sin(\theta_1 + \theta_2) \hat{n}_y] \end{align*} \] --- ### **4. Combine \( \hat{n}_x \) and \( \hat{n}_y \) Terms:** **For \( \hat{n}_x \):** \[ L_1 + L_2\cos\theta_1 + L_3\cos(\theta_1 + \theta_2) \] **For \( \hat{n}_y \):** \[ L_2\sin\theta_1 + L_3\sin(\theta_1 + \theta_2) \] --- ## **Final Answer** \[ \boxed{ \vec{r}_{R/O} = \left[ L_1 + L_2\cos\theta_1 + L_3\cos(\theta_1 + \theta_2)\right] \hat{n}_x + \left[ L_2\sin\theta_1 + L_3\sin(\theta_1 + \theta_2)\right] \hat{n}_y } \] --- ### **Summary Table** | Link | \( x \)-component | \( y \)-component | |-------------------|------------------------------------------------|------------------------------------------------| | \( O \to P \) | \( L_1 \) | | | \( P \to Q \) | \( L_2\cos\theta_1 \) | \( L_2\sin\theta_1 \) | | \( Q \to R \) | \( L_3\cos(\theta_1 + \theta_2) \) | \( L_3\sin(\theta_1 + \theta_2) \) | --- ## **Let me know if you need a step-by-step diagram or further explanation on any part!**

# Detailed Solution for Problem 3 ## **Overview** In this problem, we need to find the position vector of point \( R \) with respect to the origin \( O \) for a coordinate measuring machine with known arm lengths \( L_1, L_2, L_3 \) and variable angles \( \theta_1 \) and \( \theta_2 \). ### **Required Form** The answer should be in the form: \[ \vec{r}_{R/O} = (x) \hat{n}_x + (y) \hat{n}_y \] ## **Step 1: Analyze the Structure** 1. **Link from \( O \) to \( P \)**: - Length: \( L_1 \) - Direction: Along \( \hat{n}_x \) 2. **Link from \( P \) to \( Q \)**: - Length: \( L_2 \) - Direction: At angle \( \theta_1 \) 3. **Link from \( Q \) to \( R \)**: - Length: \( L_3 \) - Direction: At angle \( \theta_1 + \theta_2 \) ## **Step 2: Write Position Vectors** ### **1. From \( O \) to \( P \)** \[ \vec{r}_{P/O} = L_1 \hat{n}_x \] ### **2. From \( P \) to \( Q \)** \[ \vec{r}_{Q/P} = L_2 \left( \cos \theta_1 \hat{n}_x + \sin \theta_1 \hat{n}_y \right) \] ### **3. From \( Q \) to \( R \)** \[ \vec{r}_{R/Q} = L_3 \left( \cos(\theta_1 + \theta_2) \hat{n}_x + \sin(\theta_1 + \theta_2) \hat{n}_y \right) \] ## **Step 3: Combine the Vectors** To find the total position vector \( \vec{r}_{R/O} \), we sum all the position vectors: \[ \vec{r}_{R/O} = \vec{r}_{P/O} + \vec{r}_{Q/P} + \vec{r}_{R/Q} \] Substituting the components: \[ \begin{align*} \vec{r}_{R/O} = &\ L_1 \hat{n}_x \\ &+ L_2 \left( \cos \theta_1 \hat{n}_x + \sin \theta_1 \hat{n}_y \right) \\ &+ L_3 \left( \cos(\theta_1 + \theta_2) \hat{n}_x + \sin(\theta_1 + \theta_2) \hat{n}_y \right) \end{align*} \] ## **Step 4: Rearranging Terms** ### **Combine \( \hat{n}_x \) Components:** \[ x = L_1 + L_2 \cos \theta_1 + L_3 \cos(\theta_1 + \theta_2) \] ### **Combine \( \hat{n}_y \) Components:** \[ y = L_2 \sin \theta_1 + L_3 \sin(\theta_1 + \theta_2) \] ## **Final Position Vector** Combining the components, we have: \[ \vec{r}_{R/O} = \left( L_1 + L_2 \cos \theta_1 + L_3 \cos(\theta_1 + \theta_2) \right) \hat{n}_x + \left( L_2 \sin \theta_1 + L_3 \sin(\theta_1 + \theta_2) \right) \hat{n}_y \] ## **Summary** The final position vector denoting the tip \( R \) with respect to the origin \( O \) is: \[ \boxed{ \vec{r}_{R/O} = \left( L_1 + L_2 \cos \theta_1 + L_3 \cos(\theta_1 + \theta_2) \right) \hat{n}_x + \left( L_2 \sin \theta_1 + L_3 \sin(\theta_1 + \theta_2) \right) \hat{n}_y } \] This vector describes the location of point \( R \) based on the lengths of the arms and angles of rotation. Let me know if you need further clarification or additional examples!