If  θ is the angle between two vectors \(\vec{a}\)  and \(\vec{b}\) then \(\vec{a}\).\(\vec{b}\) ≥0 only when- 

0 ≤θ ≤ π/2

Let θ is the angle between two vectors \(\vec{a}\)  and \(\vec{b}\) then \(\vec{a}\) . \(\vec{b}\)≥0

Then, without loss of generality, \(\vec{a}\) and \(\vec{b}\)are non-zero vectors so that |\(\vec{a}\)| and | \(\vec{b}\) | are positive.

it is known that (\(\vec{a}\) .\(\vec{b}\)) =|\(\vec{a}\) |.|\(\vec{b}\) | cosθ

⇒ | \(\vec{a}\)|.|\(\vec{b}\) |cosθ ≥0

⇒ cosθ ≥0

⇒ 0 ≤ θ ≤ π/2

Hence \(\vec{a}\).\(\vec{b}\)≥0 when  0 ≤θ ≤ π/2