## Step-by-Step Solution for Determining Miller Indices of Directions ### **Step 1: Understanding Miller Indices for Directions** The Miller indices for directions in a crystal are represented as **[uvw]**. For a vector from point \( P_1 (x_1, y_1, z_1) \) to \( P_2 (x_2, y_2, z_2) \), you: - Subtract coordinates: \( (u, v, w) = (x_2-x_1, y_2-y_1, z_2-z_1) \) - Reduce to smallest integer values if necessary. ### **Step 2: Identify the Red Arrows' Start and End Points** Let's analyze each arrow in the image. #### **Arrow 1 (Bottom left to top left)** - **Start:** (, , ) - **End:** (, , 1) - **Direction:** \( (, , 1) \) - **Miller Index:** \([001]\) #### **Arrow 2 (Bottom left to bottom right)** - **Start:** (, , ) - **End:** (1, , ) - **Direction:** \( (1, , ) \) - **Miller Index:** \([100]\) #### **Arrow 3 (Bottom left to "b" point)** - **Start:** (, , ) - **End:** (, 1, ) - **Direction:** \( (, 1, ) \) - **Miller Index:** \([010]\) #### **Arrow 4 (Bottom left to top right)** - **Start:** (, , ) - **End:** (1, 1, 1) - **Direction:** \( (1, 1, 1) \) - **Miller Index:** \([111]\) #### **Arrow 5 (Middle left to top middle)** - **Start:** (, 1, ) - **End:** (, 1, 1) - **Direction:** \( (, , 1) \) - **Miller Index:** \([001]\) #### **Arrow 6 (Bottom left to middle top)** - **Start:** (, , ) - **End:** (, 1, 1) - **Direction:** \( (, 1, 1) \) - **Miller Index:** \([011]\) ### **Step 3: List All Miller Indices** | Arrow | Direction Vector | Miller Index | |-------|------------------|-------------| | 1 | (, , 1) | [001] | | 2 | (1, , ) | [100] | | 3 | (, 1, ) | [010] | | 4 | (1, 1, 1) | [111] | | 5 | (, , 1) | [001] | | 6 | (, 1, 1) | [011] | --- ## **Final Answers (Miller Indices for Red Arrow Directions)** \[ \boxed{ \begin{align*} 1. & \quad [001] \\ 2. & \quad [100] \\ 3. & \quad [010] \\ 4. & \quad [111] \\ 5. & \quad [001] \\ 6. & \quad [011] \\ \end{align*} } \] --- **Explanation:** - Each Miller index is derived by calculating the vector components in terms of lattice constants \(a, b, c\). - The answer is given in standard [uvw] form for directions.

# Step-by-Step Solution for Determining Miller Indices of Directions ## Step 1: Understanding Miller Indices for Directions Miller indices for directions in a crystal are expressed as **[uvw]**. For a vector from point \( P_1 (x_1, y_1, z_1) \) to \( P_2 (x_2, y_2, z_2) \), follow these steps: - Calculate the direction vector: \[ (u, v, w) = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \] - Reduce the components to the smallest integer values. ## Step 2: Identify the Red Arrows' Start and End Points Analyze each red arrow in the image for their start and end points. ### Arrow 1 (Bottom left to top left) - **Start:** \( (0, 0, 0) \) - **End:** \( (0, 0, 1) \) - **Direction Vector:** \( (0, 0, 1) \) - **Miller Index:** \([001]\) ### Arrow 2 (Bottom left to bottom right) - **Start:** \( (0, 0, 0) \) - **End:** \( (1, 0, 0) \) - **Direction Vector:** \( (1, 0, 0) \) - **Miller Index:** \([100]\) ### Arrow 3 (Bottom left to "b" point) - **Start:** \( (0, 0, 0) \) - **End:** \( (0, 1, 0) \) - **Direction Vector:** \( (0, 1, 0) \) - **Miller Index:** \([010]\) ### Arrow 4 (Bottom left to top right) - **Start:** \( (0, 0, 0) \) - **End:** \( (1, 1, 1) \) - **Direction Vector:** \( (1, 1, 1) \) - **Miller Index:** \([111]\) ### Arrow 5 (Middle left to top middle) - **Start:** \( (0, 1, 0) \) - **End:** \( (0, 1, 1) \) - **Direction Vector:** \( (0, 0, 1) \) - **Miller Index:** \([001]\) ### Arrow 6 (Bottom left to middle top) - **Start:** \( (0, 0, 0) \) - **End:** \( (0, 1, 1) \) - **Direction Vector:** \( (0, 1, 1) \) - **Miller Index:** \([011]\) ## Step 3: List All Miller Indices | Arrow | Direction Vector | Miller Index | |-------|------------------|--------------| | 1 | (0, 0, 1) | [001] | | 2 | (1, 0, 0) | [100] | | 3 | (0, 1, 0) | [010] | | 4 | (1, 1, 1) | [111] | | 5 | (0, 0, 1) | [001] | | 6 | (0, 1, 1) | [011] | --- ## Final Answers (Miller Indices for Red Arrow Directions) \[ \boxed{ \begin{align*} 1. & \quad [001] \\ 2. & \quad [100] \\ 3. & \quad [010] \\ 4. & \quad [111] \\ 5. & \quad [001] \\ 6. & \quad [011] \\ \end{align*} } \] --- **Explanation:** Each Miller index represents the direction of the arrows in terms of the crystal lattice, normalized to the smallest integer values.