Question

Let `overlinea, overlineb` and `overlinec` be three vectors having magnitudes 1, 1, and 2 respectively. If `overlinea xx (overlinea xx overlinec) + overlineb = overline0`, then the acute angle between `overlinea` and `overlinec` is ______ 

विकल्प

• `pi/4`

• `pi/6`

• `pi/3`

• none of these

Answer 1

Let `overlinea, overlineb` and `overlinec` be three vectors having magnitudes 1, 1, and 2 respectively. If `overlinea xx (overlinea xx overlinec) + overlineb = overline0`, then the acute angle between `overlinea` and `overlinec` is `underline(pi/6)`.

Explanation:

Given, `|overlinea| = 1, |overlineb| = 1` and `|overlinec| = 2`

Also, `overlinea xx (overlinea xx overlinec) + overlineb = overline0`

⇒ `(overlinea . overlinec)overlinea - (overlinea.overlinea)overlinec + overlineb = overline0`

⇒ `(overlinea.overlinec)overlinea - overlinec + overlineb = overline0` .................`[∵ overlinea.overlinea = |overlinea|^2 = 1]`

⇒ `(overlinea.overlinec)overlinea - overlinec = -overlineb`

⇒ `|(overlinea.overlinec)overlinea - overlinec| = |-overlineb|`

⇒ `|(overlinea.overlinec)overlinea - overlinec|^2 = |overlineb|^2`

⇒ `|(overlinea.overlinec)overlinea|^2 + |overlinec|^2 = 2{(overlinea.overlinec)overlinea.overlinec} = |overlineb|^2`

⇒ `(overlinea.overlinec)^2|overlinea|^2 + |overlinec|^2 - 2(overlinea.overlinec)(overlinea.overlinec) = |overlineb|^2`

⇒ `(overlinea.overlinec)^2{|overlinea|^2 - 2} + |overlinec|^2 = |overlineb|^2`

⇒ `-(overlinea.overlinec)^2 + 4 = 1`  ...............`[|overlineb|^2 = 1, |overlinec|^2 = 4]`

⇒ `(overlinea.overlinec)^2 = 3`

⇒ `overlinea.overlinec = ±sqrt3`

⇒ `|overlinea||overlinec| costheta = sqrt3`,

where θ is an acute angle between `overlinea` and `overlinec`

⇒ cos θ = `sqrt3/2 ⇒ theta = pi/6`