Question

If θ is the angle between two vectors $\vec a \ and \ \vec b$, then $\vec a \ . \vec b \geq 0$ only when
(A) 0< θ< $\frac{\pi}{2}$
(A) 0≤ θ≤ $\frac{\pi}{2}$
(A) 0< θ< $\pi$
(A) 0≤ θ≤ $\pi$

Answer 1

Given, There are two vectors a and b, such that there dot product is greater than equal to zero.

Now, |a.b| = |a||b|Cosθ

It is given that a.b is greater than equal to zero, therefore,

|a||b|Cosθ ≥ 0

Now, Magnitude of a and b always have the value greater than 1. Therefore, everything of positive depends on Cosθ.

∴ Cosθ ≥ 0

Now, Cosθ is positive in the range of 0 ≤ θ ≤ π/2.

Cosθ is positive in Ist Quadrant and also positive in Fourth last quadrant. There range is,

  3π/4 < θ < 2π

Now, there is only range of Ist Quadrant.

Hence, Option (B). is correct.

Option (A). is not correct because, Cosine have a value of 1 at zero degree which greater than 1, thus there should be sign of ≤.

Hope it helps.