3D Geometry
Angle between Line and Plane
Grade 12
Question:
<p>The angle between the line $\frac{x + 1}{2} = \frac{y}{3} = \frac{z - 3}{6}$ and the plane $10x + 2y - 11z = 3$ is</p>
<p>(a) $\frac{\pi}{2}$</p>
<p>(b) $\frac{\pi}{4}$</p>
<p>(c) $\frac{\pi}{6}$</p>
<p>(d) $\sin^{-1}\left(\frac{8}{21}\right)$</p>
Step-by-Step Solution
Key Concept: The angle between a line and a plane is the complement of the angle between the line and the normal to the plane. Use the dot product formula with direction ratios.
Solution: Let $\theta$ be the angle between the line and the plane. From the line equation: $a = 2$, $b = 3$, $c = 6$ From the plane equation: $a_1 = 10$, $b_1 = 2$, $c_1 = -11$ Using the formula: $\sin\theta = \frac{|aa_1 + bb_1 + cc_1|}{\sqrt{a^2 + b^2 + c^2} \cdot \sqrt{a_1^2 + b_1^2 + c_1^2}}$ $= \frac{|(2)(10) + (3)(2) + (6)(-11)|}{\sqrt{2^2 + 3^2 + 6^2} \cdot \sqrt{10^2 + 2^2 + 11^2}}$ $= \frac{|20 + 6 - 66|}{\sqrt{49} \cdot \sqrt{225}}$ $= \frac{|-40|}{7 \times 15} = \frac{40}{105} = \frac{8}{21}$ Therefore, $\theta = \sin^{-1}\left(\frac{8}{21}\right)$
Correct Answer: D