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If the angle between the line x = (y - 1)/2 = (z - 3)/λ and the plane x + 2y + 3z = 4 is cos⁻¹(√(5/14)), then λ equals

1. 3/2
2. 2/5
3. 5/3
4. 2/3

Correct answer: 2/3

Solution

Line direction d=(1,2,lambda), plane normal n=(1,2,3). The angle has cos = sqrt(5/14), so sin^2 = 9/14. Then (d.n)^2/(|d|^2|n|^2) = (5+3lambda)^2/((5+lambda^2)*14) = 9/14, which solves to lambda = 2/3.

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