Question

Find the angle between the two planes \( 2x + y - 2z = 5 \) and \( 3x - 6y - 2z = 7 \).

• A. \( \cos^{-1} \left( \frac{4}{21} \right) \)
• B. \( \cos^{-1} \left( \frac{2}{11} \right) \)
• C. \( \cos^{-1} \left( \frac{2}{21} \right) \)
• D. \( \cos^{-1} \left( \frac{1}{11} \right) \)

Hint:

Use the formula for the angle between two planes to find the cosine of the angle and then use the inverse cosine to get the angle.

Answer 1

The Correct Option is A

Step 1: Use the formula for the angle between two planes.
The formula for the angle between two planes is: \[ \cos \theta = \frac{|A_1 A_2 + B_1 B_2 + C_1 C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2} \sqrt{A_2^2 + B_2^2 + C_2^2}} \] where \( A_1, B_1, C_1 \) are the coefficients of the first plane and \( A_2, B_2, C_2 \) are the coefficients of the second plane.
Step 2: Conclusion.
After substituting the values, we get \( \cos^{-1} \left( \frac{4}{21} \right) \). Final Answer: \[ \boxed{\cos^{-1} \left( \frac{4}{21} \right)} \]