Let ` vec a` and `vec b` be two unit vectors and `alpha` be the angle between them, then ` vec a+ vec b` is a unit vectors, if

Options

A. `alpha = (pi)/(4)`
B. `alpha = (pi)/(3)`
C. `alpha = (pi)/(2)`
D. `alpha = (2pi)/(3)`

Answer

D

Explanation

Let `vec(a)` and `vec(b)` are two unit vectors and angle between them is `theta`.
then, `|vec(a)|=|vec(b)| = 1`
Now, `(vec(a) + vec(b))` is a unit vector if `|vec(a) + vec(b)|=1`
`implies (vec(a) + vec(b))^(2)=1`
`implies (vec(a)+ vec(b)). (vec(a) + vec(b)) = 1`
`implies vec(a).vec(a) + vec(a).vec(b) + vec(b) . vec(b).vec(a) + vec(b) .vec(b) = 1`
`implies |vec(a)|^(2) + |vec(b)|^(2) + 2vec(a).vec(b) =1 (because a.b = b.a)`
`implies 1^(2) + 1^(2) + 2vec(a).vec(b) = 1`
`implies vec(a).vec(b) =- (1)/(2)`
`implies |vec(a)||vec(b)|cos theta =- (1)/(2)`
`implies 1xx1xxcos theta = - (1)/(2)`
`implies cos theta =- (1)/(2) implies theta = (2pi)/(3)`