Let's break down and solve **Problem 1** step by step. --- # Problem 1: Find \( R_{A/E} \) and Confirm it's a Rotation Matrix ## Step 1: Understanding the Problem - **\( R_{A/E} \)** represents a rotation matrix from frame \( E \) (Earth) to frame \( A \). - We are **not given explicit vectors** for the axes, so typically, \( R_{A/E} \) is a 3×3 matrix whose columns are the unit vectors of Frame \( A \) expressed in Frame \( E \). - The matrix must satisfy two conditions: 1. **Orthogonality:** \( R_{A/E}^T R_{A/E} = I \) 2. **Determinant:** \( \det(R_{A/E}) = 1 \) Since the problem does **not give numerical data**, let's **outline the general procedure** (as often expected in such problems): --- ## Step 2: Constructing the Rotation Matrix Assume the axes of frame \( A \) expressed in \( E \) are: - \(\mathbf{i}_A\) - \(\mathbf{j}_A\) - \(\mathbf{k}_A\) and their components in \( E \) are: \[ \mathbf{i}_A = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}_E,\quad \mathbf{j}_A = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}_E,\quad \mathbf{k}_A = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix}_E \] The rotation matrix is: \[ R_{A/E} = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} \] --- ## Step 3: Check Rotation Matrix Properties 1. **Orthogonality:** - The columns must be **mutually perpendicular** and **unit length**. - \( R_{A/E}^T R_{A/E} = I \) 2. **Determinant:** - \( \det(R_{A/E}) = 1 \) --- ## Step 4: Example with Numbers (for demonstration) Suppose you have: \[ \mathbf{i}_A = \begin{bmatrix} 1 \\ \\ \end{bmatrix}_E,\quad \mathbf{j}_A = \begin{bmatrix} \\ 1 \\ \end{bmatrix}_E,\quad \mathbf{k}_A = \begin{bmatrix} \\ \\ 1 \end{bmatrix}_E \] \[ R_{A/E} = \begin{bmatrix} 1 & & \\ & 1 & \\ & & 1 \end{bmatrix} \] - \( R_{A/E}^T R_{A/E} = I \) - \( \det(R_{A/E}) = 1 \) This is the identity matrix—a valid rotation matrix. --- ## Step 5: Final Answer (General Form) \[ \boxed{ R_{A/E} = \begin{bmatrix} \mathbf{i}_A & \mathbf{j}_A & \mathbf{k}_A \end{bmatrix}_E } \] - Columns: unit vectors of Frame \( A \) in Frame \( E \) - Must be orthonormal and have determinant 1. --- ### **Summary Table** | Step | Action | Example/Formula | |------|----------------------------------------|-------------------------------| | 1 | Write axes of Frame \( A \) in \( E \) | \( \mathbf{i}_A, \mathbf{j}_A, \mathbf{k}_A \) | | 2 | Form matrix with these as columns | \( R_{A/E} = [\mathbf{i}_A \ \mathbf{j}_A \ \mathbf{k}_A] \) | | 3 | Check orthogonality and determinant | \( R^T R = I, \det R = 1 \) | --- If you provide specific vectors or points, I can compute the matrix in fraction form. Otherwise, the above **is the step-by-step general solution** as requested. --- Would you like a concrete example using specific vectors or are you looking for the **general method** as above?