Absolutely! Let's solve the Miller indices step by step, just like I would explain it to you in person: --- **Step 1: Write the intercepts** - The plane intercepts the axes at: - \( x = \frac{a}{2} \) (at point p) - \( y = a \) (at point r) - \( z = \frac{2a}{3} \) (at point q) --- **Step 2: Express intercepts in terms of 'a'** - For x: \( \frac{a}{2} \Rightarrow \frac{1}{2}a \) - For y: \( a \Rightarrow 1a \) - For z: \( \frac{2a}{3} \Rightarrow \frac{2}{3}a \) --- **Step 3: Take reciprocals** - x-intercept reciprocal: \( \frac{1}{\frac{1}{2}} = 2 \) - y-intercept reciprocal: \( \frac{1}{1} = 1 \) - z-intercept reciprocal: \( \frac{1}{\frac{2}{3}} = \frac{3}{2} \) --- **Step 4: Clear fractions (multiply all by 2 to get whole numbers)** - x: \( 2 \times 2 = 4 \) - y: \( 1 \times 2 = 2 \) - z: \( \frac{3}{2} \times 2 = 3 \) --- **Step 5: Write Miller indices in parentheses** - Miller indices are **(4 2 3)** --- **Final Answer:** **The Miller indices for the plane are (4 2 3).** Let me know if you want a recap or have questions!

Great! Let's go through this step-by-step, just like I would explain to you in class. --- **Given intercepts:** - x-intercept at \( p \): \( \frac{a}{2} \) - y-intercept at \( r \): \( \frac{3a}{4} \) - z-intercept at \( q \): \( \frac{2a}{3} \) --- **Step 1: Write the intercepts** - \( x = \frac{a}{2} \) - \( y = \frac{3a}{4} \) - \( z = \frac{2a}{3} \) --- **Step 2: Express in terms of 'a'** - x: \( \frac{a}{2} \) - y: \( \frac{3a}{4} \) - z: \( \frac{2a}{3} \) --- **Step 3: Take reciprocals** - x: \( \frac{1}{\frac{a}{2}} = \frac{2}{a} \) - y: \( \frac{1}{\frac{3a}{4}} = \frac{4}{3a} \) - z: \( \frac{1}{\frac{2a}{3}} = \frac{3}{2a} \) Since all are divided by 'a', to simplify, focus on the reciprocals relative to the intercepts: - x: \( \frac{1}{\frac{a}{2}} = \frac{2}{a} \) - y: \( \frac{1}{\frac{3a}{4}} = \frac{4}{3a} \) - z: \( \frac{1}{\frac{2a}{3}} = \frac{3}{2a} \) But to find Miller indices, we take the reciprocals of intercepts in terms of 'a', ignoring 'a' itself (since it cancels out): - x: \( \frac{1}{\frac{1}{2}} = 2 \) - y: \( \frac{1}{\frac{3}{4}} = \frac{4}{3} \) - z: \( \frac{1}{\frac{2}{3}} = \frac{3}{2} \) --- **Step 4: Clear fractions** Find the least common multiple (LCM) of denominators: 3 and 2. - Multiply all by 6: - x: \( 2 \times 6 = 12 \) - y: \( \frac{4}{3} \times 6 = 8 \) - z: \( \frac{3}{2} \times 6 = 9 \) --- **Step 5: Final Miller indices** - The Miller indices are **(12 8 9)**. --- **Answer:** **The Miller indices are (12 8 9).** Let me know if you'd like to review any step!