Question

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

• A. 0<θ<\(\frac{\pi}{2}\)
• B. 0≤θ≤\(\frac{\pi}{2}\)
• C. 0<θ<π
• D. 0≤θ≤π

Answer 1

The Correct Option is B

Let θ be the angle between two vectors \(\vec a\) and \(\vec b\).
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.
∴\(\vec a\).\(\vec b\)≥0
⇒ |\(\vec a\)||\(\vec b\)|cosθ≥0
⇒ cosθ≥0 [|\(\vec a\)|and|\(\vec b\)|are positive]
⇒ 0≤θ≤\(\frac{\pi}{2}\)
Hence,\(\vec a\).\(\vec b\)≥0 when 0≤θ≤\(\frac{\pi}{2}\).
The correct answer is B.