Question

Find the angle \( \theta \) between the line \( \frac{x-1}{1} = \frac{y+2}{-2} = \frac{z-3}{2} \) and the plane \( 2x - y + 2z = 7 \).

• A. \( \sin^{-1}\left(\frac{8}{9}\right) \)
• B. \( \cos^{-1}\left(\frac{8}{9}\right) \)
• C. \( \frac{\pi}{2} \)
• D. \( \sin^{-1}\left(\frac{2}{3}\right) \)

Hint:

Remember the unique rule for mixed line-and-plane problems: finding the angle between a line and a plane uses \( \sin\theta \), whereas finding the angle between two lines or two planes uses \( \cos\theta \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The angle between a line and a plane is not calculated the same way as the angle between two lines.
If \(\alpha\) is the angle between the line's direction vector and the plane's normal vector, then the angle \(\theta\) between the line and the plane is the complement: \(\theta = 90^{\circ} - \alpha\).
Therefore, while we use \(\cos \alpha\) for vectors, we use \(\sin \theta\) for the line-plane relationship because \(\cos(90^{\circ} - \theta) = \sin \theta\).
Step 2: Key Formula or Approach:
The formula for the angle \(\theta\) is:
\[ \sin \theta = \left| \frac{\vec{b} \cdot \vec{n}}{|\vec{b}| |\vec{n}|} \right| \]
where \(\vec{b}\) is the direction vector of the line and \(\vec{n}\) is the normal vector to the plane.
Step 3: Detailed Explanation:
Extract the direction vector \(\vec{b}\) from the denominators of the symmetric line equation:
\(\vec{b} = 1\hat{i} - 2\hat{j} + 2\hat{k}\).
Extract the normal vector \(\vec{n}\) from the coefficients of \(x, y, z\) in the plane equation:
\(\vec{n} = 2\hat{i} - 1\hat{j} + 2\hat{k}\).
Calculate the scalar dot product \(\vec{b} \cdot \vec{n}\):
\[ \vec{b} \cdot \vec{n} = (1)(2) + (-2)(-1) + (2)(2) = 2 + 2 + 4 = 8 \]
Calculate the magnitude of vector \(\vec{b}\):
\[ |\vec{b}| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]
Calculate the magnitude of normal vector \(\vec{n}\):
\[ |\vec{n}| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]
Substitute the calculated values into the \(\sin \theta\) formula:
\[ \sin \theta = \frac{8}{3 \times 3} = \frac{8}{9} \]
Isolating \(\theta\):
\[ \theta = \sin^{-1} \left(\frac{8}{9}\right) \]
Step 4: Final Answer:
The angle is \(\sin^{-1} \left(\frac{8}{9}\right)\), matching option (A).