Question

The angle between the line r = `(hat"i" + hat"j" - hat"k") + λ (3hat"i" + hat"j")` and the plane `"r". (hat"i" + 2hat"j" + 3hat"k")` = 8 is ____________.

Options

• `sin^-1 ((2sqrt7)/sqrt5)`

• `sin^-1 (sqrt5/(2sqrt7))`

• `sin^-1 ((3sqrt7)/sqrt5)`

• `sin^-1 (sqrt7/(3sqrt5))`

Answer 1

The angle between the line r = `(hat"i" + hat"j" - hat"k") + λ (3hat"i" + hat"j")` and the plane `"r". (hat"i" + 2hat"j" + 3hat"k")` = 8 is `underline(sin^-1 (sqrt5/(2sqrt7)))`.

Explanation:

Given, line r = `(hat"i" + hat"j" - hat"k") + λ (3hat"i" + hat"j")`

and plane = `"r". (hat"i" + 2hat"j" + 3hat"k")` = 8

Here, b = `3hat"i" + hat"j"` and n = `hat"i" + 2hat"j" + 3hat"k"`

Angle between line and plane is

sin θ = `("b"."n")/(|"b"| |"n"|)`

sin θ = `((3hat"i" + hat"j").(hat"i" + 2hat"j" + 3hat"k"))/(sqrt(9 + 1) sqrt(1 + 4 + 9)) = (3 + 2)/(sqrt10 sqrt14)`

sin θ = `5/(2sqrt5 sqrt7) = sqrt5/(2sqrt7)`

∵ θ = `sin^-1 (sqrt5/(2sqrt7))`