Question

Solve the following :

Find the angle between the planes `bar"r".(-2hat"i" + hat"j" + 2hat"k")` = 17 and `bar"r".(2hat"i" + 2hat"j" + hat"k")` = 71.

Answer 1

The acute angle between the planes

`bar"r".bar"n"_1 = "d"_1 and bar"r".bar"n"_2 = "d"_2` is given by

`cosθ = |(bar"n"_1.bar"n"_2)/(|bar"n"_1||bar"n"_2|)|`     ...(1)
Here,
`bar"n"_1 = -2hat"i" + hat"j" + 2hat"k"`,

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

∴  `=|((-2hat"i" + hat"j" + 2hat"k").(2hat"i" + 2hat"j" + hat"k"))/(sqrt((-2)^2+1^2+2^2)sqrt(2^2+2^2+1^2))|` 

`=|(-4+2+2)/(3xx3)| = |0/9|`

Cosθ = 0

∴ θ = 90°.