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) $0\le\theta\le\pi$

> [!TLDR] > The dot product $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$ is non-negative when $\cos\theta \ge 0$, which occurs for $0 \le \theta \le \frac{\pi}{2}$. The correct option is **(B)**. 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: $$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$$ 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 $\cos\theta \ge 0$?** …

If θ is the angle between two vectors veca and vec

Concept understanding — Dot Product Sign

The Dot Product Sign: What Does It Tell You?

Imagine pushing a heavy box across the floor at an angle — partly forward, partly sideways. The dot product tells you how much of your effort actually moves the box forward.

Its sign — positive, negative, or zero — is the simplest possible answer to one question: Are these two vectors working together or against each other?

The Intuition First

Take two arrows (vectors) and place their tails together. The sign depends entirely on the angle between them:

Positive → the vectors point in roughly the same direction (angle less than $90^\circ$ ). They reinforce each other.
Zero → the vectors are perpendicular (exactly $90^\circ$ ). One has no effect in the direction of the other. …

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Concept understanding — Dot Product Sign

The Dot Product Sign: What Does It Tell You?

Imagine pushing a heavy box across the floor at an angle — partly forward, partly sideways. The dot product tells you how much of your effort actually moves the box forward.

Its sign — positive, negative, or zero — is the simplest possible answer to one question: Are these two vectors working together or against each other?

The Intuition First

Take two arrows (vectors) and place their tails together. The sign depends entirely on the angle between them:

Positive → the vectors point in roughly the same direction (angle less than $90^\circ$ ). They reinforce each other.
Zero → the vectors are perpendicular (exactly $90^\circ$ ). One has no effect in the direction of the other. …

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 understanding — Dot Product Sign

The Dot Product Sign: What Does It Tell You?

The Intuition First

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 understanding — Dot Product Sign

The Dot Product Sign: What Does It Tell You?

The Intuition First

Concept understanding — Dot Product Sign

The Dot Product Sign: What Does It Tell You?

The Intuition First

🔒Unlock the full solution

Concept understanding — Dot Product Sign

The Dot Product Sign: What Does It Tell You?

The Intuition First

🔒Unlock the full solution

Unlock the full solution

Unlock the full solution