Question

The angle between the two planes x + y + 2z = 9 and 2x – y + z = 15 is ______.

पर्याय

• `π/2`

• `π/3`

• π

• `(3π)/4`

Answer 1

The angle between the two planes x + y + 2z = 9 and 2x – y + z = 15 is `underlinebb(π/3)`.

Explanation:

Given planes are

x + y + 2z = 9   ...(i)

2x – y + z = 15   ...(ii)

The normal vector of first plane is

`vec(n_1) = hati + hatj + 2hatk`

And normal vector of second plane is

`vec(n_2) = 2hati - hatj + hatk`

Angle between them is

`cos θ = (vec(n_1) * vec(n_2))/(|vec(n_1)||vec(n_2)|)`

`cos θ = ((hati + hatj + 2hatk) * (2hati - hatj + hatk))/(sqrt(1^2 + 1^2 + 2^2) sqrt(2^2 + (-1)^2 + 1^2))`

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

= `3/6`

= `1/2`

`cos θ = 1/2`

`θ = cos^-1 (1/2)`

⇒ `θ = cos^-1 (cos  π/3)`

`θ = π/3`