Question

If `veca` and `vecb` are unit vectors enclosing an angle θ and `|veca + vecb| < 1`, then find the values between which θ lies.

Answer 1

`|veca + vecb| < 1`

⇒ `|veca + vecb|^2 < 1`

⇒ `|veca|^2 + |vecb|^2 + 2veca.vecb < 1`

⇒ `1 + 1 + 2veca.vecb < 1`

⇒ `veca * vecb < -1/2`

⇒ `|veca||vecb| cos θ < - 1/2`

⇒ `cos θ < - 1/2`

⇒ `-1 ≤ cos θ < - 1/2`

⇒ `(2π)/3 < θ ≤ π` i.e., `θ ∈ ((2π)/3, π]`