Question

If the angle between the planes `overliner . (mhati - hatj + 2hatk) + 3 = 0` and `overliner . (2hati - mhatj - hatk) - 5 = 0` is `pi/3`, then m = ______ 

पर्याय

• 2

• ±3

• 3

• -2

Answer 1

If the angle between the planes `overliner . (mhati - hatj + 2hatk) + 3 = 0` and `overliner . (2hati - mhatj - hatk) - 5 = 0` is `pi/3`, then m = 3.

Explanation:

`overliner . (mhati - hatj + 2hatk) + 3 = 0 ⇒ overliner . (mhati - hatj + 2hatk) = -3`

`overliner . (2hati - mhatj - hatk) - 5 = 0 ⇒ overliner . (2hati - mhatj - hatk) = 5`

Here, `overlinen_1 = mhati - hatj + 2hatk` and `overlinen_2 = 2hati - mhatj - hatk`

∴ `costheta = |(overlinen_1 . overlinen_2)/(|overlinen_1| |overlinen_2|)|`

⇒ `cos  pi/3 = |((mhati - hatj + 2hatk) . (2hati - mhatj - hatk))/(sqrt(m^2 + 1 + 4) sqrt(4 + m^2 + 1))|`

⇒ `1/2 = (2m + m - 2)/(m^2 + 5)` ........(Cosidering positive value)

⇒ m² + 5 = 6m - 4

⇒ m² - 6m + 9 = 0

⇒ (m - 3)² = 0

⇒ m = 3