Find the coordinates of the point where the line intersect the plane x – y + z – 5 = 0. Also, find the angle between the line and the plane.[CBSE 2013]

Any point on the line is of the form,

(3k + 2, 4k – 1, 2k + 2)

If the point p(3k + 2, 4k – 1, 2k + 2) lies in the plane x – y + z – 5 = 0, we have,

(3k + 2) – (4k – 1) + (2k + 2) – 5 = 0

3k + 2 – 4k + 1 + 2k + 2 – 5 = 0

k = 0

thus, the coordinates of the point of intersection of the line and the plane are :

{3(0) + 2, 4(0) – 1, 2(0) + 2}

P(2, – 1,, 2)

Let be the angle between the line and the plane . thus

, where l, m and n are the direction ratios of the line and a, b and c are the direction ratios of the normal to the plane

Here, l = 3, m = 4, n = 2, a = 1, b = – 1, and c = 1 hence,