Q.213. Angle between a plane and a line.

Question 213.
A cave has a planar roof passing through the points (0, 0, -19), (5, 0, -20) and (0, 5, -22).
A tunnel is being bored through rock from point (0, 3, 4) in the direction - i + 2j - 20k. Find the angle between the tunnel and the cave roof in degress, correct to the nearest degree.

Answer 213.
This is a question of finding the angle between the line and a plane.
Here, direction of the line is l = (-1, 2, -20)
If the plane is defined by A(0, 0, -19), B(5, 0, -20) and C(0, 5, -22), its normal is
n
= AB (vector) x (BC (vector)
= [(5, 0, -20) - (0, 0, -19)] x [(0, 5, -22) - (5, 0, -20)]
= (5, 0, -1) x (-5, 5, -2)
= (5, 15, 25)
The direction
n = (1, 3, 5)

The angle between the line and the plane is
α
= arcsin [modulus l.n] / [(modulus l) * (modulus n)]
= arcsin l (-1, 2, -20) . (1, 3, 5) l / [ l (-1, 2, -20) l * l (1, 3, 5) l ]
= arcsin l (-1 + 6 - 100) l / [√(1^2 + 2^2 + 20^2) * √(1^2 + 3^2 + 5^2)]
= arcsin 95 / √(405 * 35)
= 53°.

Link to YA!