Question

If planes `vecr.(phati - hatj + 2hatk) + 3` = 0 and `vecr.(2hati - phatj - hatk) - 5` = 0 include angle `π/3` then the value of p is ______.

विकल्प

• 1, –3

• –1, –3

• –3

• 3

Answer 1

If planes `vecr.(phati - hatj + 2hatk) + 3` = 0 and `vecr.(2hati - phatj - hatk) - 5` = 0 include angle `π/3` then the value of p is 3.

Explanation:

We have `vecr.(phati - hatj + 2hatk) + 3` = 0 and `vecr.(2hati - phatj - hatk) - 5` = 0

Since the angle between them is `π/3`

∴ cos θ = `|(vecn_1.vecn_2)/(|vecn_1|.|vecn_2|)|`

`\implies cos  π/3 ((phati - hatj + 2hatk).(2hati - phatj - hatk))/(sqrt((p)^2 + (-1)^2 + (2)^2) sqrt((2)^2 + (-p)^2 + (-1)^2`

`\implies 1/2 = (2p + p - 2)/((sqrt(p^2 + 5))(sqrt(p^2 + 5))`

`\implies 1/2 = (3p - 2)/((p^2 + 5))`

`\implies` p² + 5 = 6p – 4

`\implies` p² – 6p + 9 = 0

`\implies` (p – 3)² = 0

`\implies` p = 3