<p>Angle between the line \(\dfrac{x+1}{1} = \dfrac{y-1}{2} = \dfrac{z-2}{2}\) and plane \(2x - y + \sqrt{\lambda}z + 4 = 0\) is given by \(\sin\theta = \dfrac{1}{3}\). Find \(\lambda = ?\)</p>
Step-by-Step Solution
Key Concept: The sine of the angle between a line and plane equals the cosine of the angle between the line's direction vector and the plane's normal vector. Use the formula: sin θ = |a·n|/(|a||n|) where a is the direction vector and n is the normal vector.
Step 1: Identify the direction vector of the line and normal to the plane. Line: a = (1, 2, 2) Plane: n = (2, -1, √λ) Step 2: Apply the sine formula for angle between line and plane. sin θ = | a · n |/(| a || n |) = 1/3 Step 3: Calculate the dot product and magnitudes. a · n = 1(2) + 2(-1) + 2(√λ) = 2 - 2 + 2√λ = 2√λ | a | = √(1^2 + 2^2 + 2^2) = √9 = 3 | n | = √(4 + 1 + λ) = √(5 + λ) Step 4: Substitute into the formula. |2√λ|/(3·√(5 + λ)) = 1/3 2√λ/(3√(5 + λ)) = 1/3 Step 5: Solve for λ. 2√λ = √(5 + λ) 4λ = 5 + λ 3λ = 5 ∴ λ = 5/3
Correct Answer: 5/3