Question

The angle between the line with the direction ratios \( (2, 5, 1) \) and the plane \( 8x + 2y - z = 4 \) is given by

• A. \( \cos^{-1} \left(\frac{64}{\sqrt{9804}}\right) \)
• B. \( \sin^{-1} \left(\frac{64}{\sqrt{9804}}\right) \)
• C. \( \sin^{-1} \left(\frac{25}{\sqrt{2070}}\right) \)
• D. \( \cos^{-1} \left(\frac{25}{\sqrt{2070}}\right) \)

Hint:

To find the angle between a line and a plane, use the normal vector of the plane and the direction vector of the line. The sine of the complementary angle to the angle calculated by the dot product gives the desired result.

Answer 1

The Correct Option is C

Step 1: Identify the normal vector of the plane \( \mathbf{n} = \langle 8, 2, -1 \rangle \) and the direction vector of the line \( \mathbf{d} = \langle 2, 5, 1 \rangle \). 
Step 2: Calculate the angle \( \theta \) between the line and the plane using the dot product: \[ \cos \theta = \frac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}| |\mathbf{n}|}. \] Substitute values: \[ \cos \theta = \frac{|2 \cdot 8 + 5 \cdot 2 - 1 \cdot 1|}{\sqrt{2^2 + 5^2 + 1^2} \cdot \sqrt{8^2 + 2^2 + (-1)^2}} = \frac{25}{\sqrt{2070}}. \] Step 3: Since the problem is about the angle between the line and the plane, we calculate the complementary angle \( \phi = 90^\circ - \theta \). Therefore, the sine function is used: \[ \sin \phi = \frac{25}{\sqrt{2070}}. \]