Question

Find the angle between the line `(x - 1)/3 = (y + 1)/2 = (z + 2)/4` and the plane 2x + y − 3z + 4 = 0.

Answer 1

The angle θ between the line

`(x - x_1)/a_1 = (y - y_1)/b_1 = (z - z_1)/c_1` and the plane ax + by + cz + d = 0 is given by 

sinθ = `(aa_1 + b b_1 + c c_1)/(sqrt(a^2 + b^2 + c^2) . sqrt(a_1^2 + b_1^2 + c_1^2))`

Here , `a_1 = 3 , b_1 = 2 , c_1 = 4   "and"  a = 2 , b = 1 , c = -3`

∴ `aa_1 + b b_1 + c c_1 = 2(3) + 1(2) + (-3)(4)`

= 6 + 2 - 12 = -4

`sqrt(a^2 + b^2 + c^2) = sqrt(2^2 + 1^2 + (-3)^2) = sqrt(4 + 1 + 9) = sqrt(14)`

and `sqrt(a_1^2 + b_1^2 + c_1^2) = sqrt(3^2 + 2^2 + 4^2) = sqrt(9 + 4 + 16) = sqrt(29)`

∴ sin θ = `(-4)/(sqrt(14) . sqrt(29)) = (-4)/sqrt(406)`

∴ `θ = sin^-1((-4)/sqrt(406))`