Let ` vec a\ a n d\ vec b` be two unit vectors and `alpha` be the angle between them, then ` vec a+ vec b` is a unit vectors, if `alpha=pi/4` b. `alpha=pi/3` c. `alpha=(2pi)/3` d. `alpha=pi/2`

Options

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

Answer

C

Explanation

Let `vec(a) and vec(b)` be two unit vectors.
`:. |a|=1 and |b|=1`
Since, `alpha` is the angle between `vec(a) and vec(b)`
`:. Cos alpha = (vec(a).vec(b))/(|a||b|)`
`cos alpha = (vec(a).vec(b))/(1)`
`cos alpha = vec(a).vec(b)`
Now, `|vec(a)+vec(b)|=1` (`:' vec(a)+vec(b)` is unit vector)
Squaring both sides
`rArr |vec(a)|^(2)+|vec(b)|^(2)+2vec(a).vec(b)=1`
`rArr 1+1+2 cos alpha = 1`
`2 cos alpha = -1`
`rArr cos alpha = -(1)/(2) = cos .(2pi)/(3)`
`rArr alpha = (2pi)/(3)`