Find the coordinates of the point where the line intersects the plane x – y + z – 5 = 0. Also find the angle between the line and the plane.
Given line
Let
⇒ x = 3r + 2, y = 4r – 1, z = 2r + 2
Substituting in the equation of the plane x – y + z – 5 = 0
We get , (3r + 2) – (4r – 1) + (2r + 2) – 5 = 0
⇒ 3r + 2 – 4r + 1 + 2r + 2 – 5 = 0
⇒ r = 0
∴ x = 3×0 + 2, y = 4×0 – 1 , z = 2×0 + 2
⇒ x = 2, y = – 1, z = 2
Direction ratios of the line are 3, 4, 2
Direction ratios of the line perpendicular to the plane are 1, – 1, 1
∴
⇒
⇒
⇒ sinθ = ⇒ θ = sin – 1(
∴ the angle between the plane and the line is sin – 1(