# Step-by-Step Solution Let's determine the Miller Indices for direction **a** in Figure 1. ## 1. **Identify the coordinates of the tail and head points of direction 'a'** From the figure: - **Tail (starting point)**: at the origin \((, , )\) - **Head (ending point)**: at the top far right corner, which, according to the cube, is at \((1, 1, 1)\) ## 2. **Write the vector for direction 'a'** The direction vector is calculated as: \[ \text{Direction vector} = (\text{head} - \text{tail}) = (1-, 1-, 1-) = (1, 1, 1) \] ## 3. **Reduce to the smallest integer values** The vector \((1, 1, 1)\) is already in the smallest integer form. ## 4. **Write the Miller Indices notation** The notation for a direction is written in square brackets: \([uvw]\), where \(u, v, w\) are the components. So, the Miller Indices for direction 'a' is: \[ [111] \] ## 5. **Check the provided options** - a. [011] - b. [101] - c. [110] - d. [101] - e. None of the above **[111] is NOT listed in the options.** --- ## **Final Answer** **e. None of the above** --- ### **Summary Table** | Step | Description | Value | |------|----------------------------|---------------| | Tail | Start point | (, , ) | | Head | End point | (1, 1, 1) | | Vector | Head - Tail | (1, 1, 1) | | Miller Indices | | [111] | | Correct Option | | e. None |