Question

Find the angle between planes `bar"r".(hat"i" + hat"j" + 2hat"k") = 13 and bar"r"(2hat"i" + hat"j" + hat"k")` = 31.

Answer 1

The acute angle θ 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 = hat"i" + hat"j" + 2hat"k"`,

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

∴ `bar"n"_1.bar"n"_2`

= `(hat"i" + hat"j" + 2hat"k").(2hat"i" - hat"j" + hat"k")`

= (1)(2) + (1)(– 1) + (2)(1)
=2 – 1 + 2
= 3
Also,
`|bar"n"_1| = sqrt(1^2 + 1^2 + 2^2) = sqrt(6)`

`|bar"n"_2| = sqrt(2^2 + (-1)^2 + 1^2) = sqrt(6)`

∴ from (1), we have

`cosθ= |(3)/(sqrt(6)sqrt(6))|`

= `(3)/(6)`

= `(1)/(2)cos60°`

∴ θ = 60°