Q.If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a}\cdot\vec{b}\ge 0$ only when (A) $0<\theta<\frac{\pi}{2}$ (B) $0\le\theta\le\frac{\pi}{2}$ (C) $0<\theta<\pi$ (D) $0\le\theta\le\pi$

Concept: Dot Product Sign — The dot product $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ is non-negative exactly when $\cos\theta \ge 0$.

Step 1: Since $|\vec{a}||\vec{b}| > 0$ for non-zero vectors, the sign depends only on $\cos\theta$.

Step 2: $\cos\theta \ge 0$ for $\theta$ in $[0, \frac{\pi}{2}]$ (including the endpoints, where $\cos 0 = 1$ and $\cos\frac{\pi}{2}=0$). …

The key to this problem lies entirely in the geometric definition of the dot product. When you see a question about the sign of $\vec{a}\cdot\vec{b}$, your first thought should be: what does the dot product tell us about the angle between the vectors?

The dot product is defined as:

where $\theta$ is the angle between the two vectors, measured from $\vec{a}$ to $\vec{b}$, and always taken between $0$ and $\pi$ (inclusive).

Since the magnitudes $|\vec{a}|$ and $|\vec{b}|$ are always non-negative (they are lengths), the sign of the dot product is entirely determined by the sign of $\cos\theta$.

1. When is $\cos\theta \ge 0$?

   The cosine function is non-negative in the first quadrant and at the boundaries. Specifically:

   • $\cos\theta = 1$ when $\theta = 0$ (vectors point in exactly the same direction)
   • $\cos\theta > 0$ when $0 < \theta < \frac{\pi}{2}$ (vectors point in generally the same direction, with an acute angle between them)
   • $\cos\theta = 0$ when $\theta = \frac{\pi}{2}$ (vectors are perpendicular)
   • $\cos\theta < 0$ when $\frac{\pi}{2} < \theta \le \pi$ (vectors point in generally opposite directions, with an obtuse angle)

2. Translating to the dot product condition

   The problem asks for $\vec{a}\cdot\vec{b} \ge 0$, which means the dot product is either positive or zero. From the above:

   • Positive dot product: $0 < \theta < \frac{\pi}{2}$
   • Zero dot product: $\theta = 0$ or $\theta = \frac{\pi}{2}$

   Combining these, the condition $\vec{a}\cdot\vec{b} \ge 0$ holds when $0 \le \theta \le \frac{\pi}{2}$. …

Method: Determining the Angle Range from the Sign of a Dot Product

Use this for questions asking for which angles $\vec a\cdot\vec b$ is positive, zero, or non-negative.

Steps

Step 1: Reduce the sign to $\cos\theta$

Since $\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta$ and the magnitudes are positive, the sign of the dot product equals the sign of $\cos\theta$.

Step 2: Find where $\cos\theta$ has the required sign …

Common Mistakes

Mistake 1: Excluding the endpoints (choosing the open interval)

Why it's wrong: $\ge0$ includes $\theta=0$ (dot product positive) and $\theta=\tfrac{\pi}{2}$ (dot product zero), so both endpoints belong. Correct approach: use the closed range $0\le\theta\le\tfrac{\pi}{2}$, not $0<\theta<\tfrac{\pi}{2}$.

Mistake 2: Confusing the $\sin$ and $\cos$ dependence …