Question

The angle between the planes and `bar"r"*(2hat"i" + hat"j" - hat"k") = 3` and `bar"r"*(hat"i" + 2hat"j" + hat"k") = 1` is ______.

Options

• `pi/2`

• `pi/3`

• `pi/4`

• `pi/6`

Answer 1

The angle between the planes and `bar"r"*(2hat"i" + hat"j" - hat"k") = 3` and `bar"r"*(hat"i" + 2hat"j" + hat"k") = 1` is `pi/3`.

Explanation:

`bar"n"_1 = 2hat"i" + hat"j" - hat"k"` and `bar"n"_2 = hat"i" + 2hat"j" + hat"k"`

cos θ = `|(bar"n"_1 * bar"n"_2)/(|bar"n"_1|*|bar"n"_2|)|`

= `|((2)(1) + (1)(2) + (-1)(1))/(sqrt(2^2 + 1^2 + (-1)^2) sqrt(1^2 + 2^2 + 1^2))|`

∴ cos θ = `|(2 + 2 - 1)/(sqrt(6) sqrt(6))| = 1/2`

⇒ θ = `pi/3`