# Explanation: Position Vector of the Tip \( R \) with Respect to Origin \( O \) Let's break down the problem step by step. ## **Understanding the Setup** - The arm consists of **three segments** with lengths \( L_1 \), \( L_2 \), and \( L_3 \). - The angles \( \theta_1 \) and \( \theta_2 \) denote the rotations at the joints. - You are to find the position vector of point \( R \) (the tip) **relative to the origin \( O \)**. - The answer should be in the form: \[ \vec{r}_{R/O} = (\text{_____})\hat{n}_x + (\text{_____})\hat{n}_y \] where \( \hat{n}_x \) and \( \hat{n}_y \) are the unit vectors along the \( x \) and \( y \) axes. --- ## **Step-by-Step Solution** ### **1. Decompose Each Segment** #### **Segment 1: \( O \) to \( P \)** - Length: \( L_1 \) - Direction: Along \( x \)-axis. - Position of \( P \) relative to \( O \): \[ \vec{r}_{P/O} = L_1 \hat{n}_x \] #### **Segment 2: \( P \) to \( Q \)** - Length: \( L_2 \) - Angle: \( \theta_1 \) from the \( x \)-axis. - The \( x \) and \( y \) components: \[ \vec{r}_{Q/P} = L_2 \cos \theta_1 \hat{n}_x + L_2 \sin \theta_1 \hat{n}_y \] #### **Segment 3: \( Q \) to \( R \)** - Length: \( L_3 \) - Angle: \( \theta_1 + \theta_2 \) from the \( x \)-axis (because the second arm rotates by \( \theta_2 \) relative to the previous segment). - The \( x \) and \( y \) components: \[ \vec{r}_{R/Q} = L_3 \cos (\theta_1 + \theta_2) \hat{n}_x + L_3 \sin (\theta_1 + \theta_2) \hat{n}_y \] --- ### **2. Add the Vectors Together** The total position vector from \( O \) to \( R \) is: \[ \vec{r}_{R/O} = \vec{r}_{P/O} + \vec{r}_{Q/P} + \vec{r}_{R/Q} \] Substitute the values: \[ \begin{align*} \vec{r}_{R/O} = \; & [L_1 \hat{n}_x] + [L_2 \cos \theta_1 \hat{n}_x + L_2 \sin \theta_1 \hat{n}_y] \\ & + [L_3 \cos (\theta_1 + \theta_2) \hat{n}_x + L_3 \sin (\theta_1 + \theta_2) \hat{n}_y] \end{align*} \] --- ### **3. Group the \( \hat{n}_x \) and \( \hat{n}_y \) Components** - \( \hat{n}_x \) component: \[ L_1 + L_2 \cos \theta_1 + L_3 \cos (\theta_1 + \theta_2) \] - \( \hat{n}_y \) component: \[ L_2 \sin \theta_1 + L_3 \sin (\theta_1 + \theta_2) \] --- ## **Final Answer** \[ \boxed{ \vec{r}_{R/O} = \left[ L_1 + L_2 \cos \theta_1 + L_3 \cos (\theta_1 + \theta_2) \right] \hat{n}_x + \left[ L_2 \sin \theta_1 + L_3 \sin (\theta_1 + \theta_2) \right] \hat{n}_y } \] **This expression gives you the position of the tip \( R \) with respect to the origin \( O \) for any values of \( L_1, L_2, L_3, \theta_1, \) and \( \theta_2 \).** --- ### **Summary Table** | Segment | \( x \)-component | \( y \)-component | |-------------|----------------------------------|-------------------------------------| | \( O \to P \) | \( L_1 \) | \( \) | | \( P \to Q \) | \( L_2 \cos \theta_1 \) | \( L_2 \sin \theta_1 \) | | \( Q \to R \) | \( L_3 \cos(\theta_1+\theta_2) \) | \( L_3 \sin(\theta_1+\theta_2) \) |

# Position Vector of the Tip \( R \) with Respect to Origin \( O \) This guide explains how to derive the position vector of point \( R \) in a 2D coordinate system based on the lengths of the arms and the angles at the joints. ## **Setup Description** - **Segments:** The arm consists of three segments with lengths \( L_1 \), \( L_2 \), and \( L_3 \). - **Angles:** The angles \( \theta_1 \) and \( \theta_2 \) are the rotations at the joints. - **Objective:** Find the position vector \( \vec{r}_{R/O} \) in the form: \[ \vec{r}_{R/O} = (x_R) \hat{n}_x + (y_R) \hat{n}_y \] --- ## **Step-by-Step Calculation** ### **1. Calculate Position of Each Segment** #### **Segment 1: From \( O \) to \( P \)** - **Length:** \( L_1 \) - **Position Vector:** \[ \vec{r}_{P/O} = L_1 \hat{n}_x \] #### **Segment 2: From \( P \) to \( Q \)** - **Length:** \( L_2 \) - **Direction:** At angle \( \theta_1 \) - **Position Vector:** \[ \vec{r}_{Q/P} = L_2 \cos \theta_1 \hat{n}_x + L_2 \sin \theta_1 \hat{n}_y \] #### **Segment 3: From \( Q \) to \( R \)** - **Length:** \( L_3 \) - **Direction:** At angle \( \theta_1 + \theta_2 \) - **Position Vector:** \[ \vec{r}_{R/Q} = L_3 \cos(\theta_1 + \theta_2) \hat{n}_x + L_3 \sin(\theta_1 + \theta_2) \hat{n}_y \] --- ### **2. Sum the Position Vectors** The total position vector from \( O \) to \( R \) is given by: \[ \vec{r}_{R/O} = \vec{r}_{P/O} + \vec{r}_{Q/P} + \vec{r}_{R/Q} \] Substituting the individual vectors: \[ \begin{align*} \vec{r}_{R/O} = & \, L_1 \hat{n}_x + \left( L_2 \cos \theta_1 \hat{n}_x + L_2 \sin \theta_1 \hat{n}_y \right) \\ & + \left( L_3 \cos(\theta_1 + \theta_2) \hat{n}_x + L_3 \sin(\theta_1 + \theta_2) \hat{n}_y \right) \end{align*} \] ### **3. Combine Components** #### **\( x \)-Component:** \[ x_R = L_1 + L_2 \cos \theta_1 + L_3 \cos(\theta_1 + \theta_2) \] #### **\( y \)-Component:** \[ y_R = L_2 \sin \theta_1 + L_3 \sin(\theta_1 + \theta_2) \] --- ## **Final Position Vector** Combining the components, the position vector of the tip \( R \) 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 expression represents the position of point \( R \) relative to the origin \( O \) based on the given lengths and angles.