Let the foot of perpendicular from a point P ( 1 , 2 , − 1 ) to the straight line L : 1 x = 0 y = − 1 z be N . Let a line be drawn from P parallel to the plane x + y + 2 z = 0 which meets L at p

Question

Let the foot of perpendicular from a point P ( 1 , 2 , − 1 ) to the straight line L : 1 x ​ = 0 y ​ = − 1 z ​ be N . Let a line be drawn from P parallel to the plane x + y + 2 z = 0 which meets L at point Q . If α is the acute angle between the lines PN and PQ , then cos α is equal to

TestHub · Accepted Answer

Let 1 x ​ = 0 y ​ = − 1 z ​ = λ ⇒ N ( λ , 0 , − λ ) b = i ^ − k ^ ∵ PN ⋅ b = 0 ⇒ 1 ( 1 − λ ) − ( λ − 1 ) = 0 ⇒ λ = 1 ⇒ N ( 1 , 0 , − 1 ) Let Q ( μ , 0 , − μ ) ∵ n = i ^ + j ^ ​ + 2 k ^ Now, ∵ PQ ​ ⋅ n = 0 ⇒ μ = − 1 ⇒ Q ( − 1 , 0 , 1 ) PN = 2 j ^ ​ and PQ ​ = 2 i ^ + 2 j ^ ​ − 2 k ^ ⇒ cos α = 3 ​ 1 ​

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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