Question

Let `overlinea = 2hati + hatj + hatk, overlineb = hati + 2hatj - hatk` and a unit vector `overlinec` be coplanar. If `overlinec` is perpendicular to `overlinea`, then `overlinec` = ______ 

विकल्प

• `1/sqrt2(-hatj + hatk)`

• `1/sqrt3(-hati - hatj - hatk)`

• `1/sqrt5(hati - 2hatj)`

• `1/sqrt3(hati - hatj - hatk)`

Answer 1

Let `overlinea = 2hati + hatj + hatk, overlineb = hati + 2hatj - hatk` and a unit vector `overlinec` be coplanar. If `overlinec` is perpendicular to `overlinea`, then `overlinec` = `underline(1/sqrt2(-hatj + hatk))`.

Explanation:

Since `overlinec` is coplanar with `overlinea` and `overlineb`.

`overlinec = xoverlinea + yoverlineb`

⇒ `overlinec = x(2hati + hatj + hatk) + y(hati + 2hatj - hatk)`

⇒ `overlinec = (2x + y)hati + (x + 2y)hatj + (x - y)hatk`

`overlinea ⊥ overlinec ⇒ overlinea . overlinec = 0`

⇒ 2(2x + y) + x + 2y + x - y = 0

⇒ y = -2x

`overlinec = -3xhatj + 3xhatk = 3x(-hatj + hatk)`

`|overlinec| = 1`

∴ `9x^2 + 9x^2 = 1`

⇒ `x = ±1/(3sqrt2) ⇒ overlinec = 1/sqrt2(-hatj + hatk)`