Q.If the vectors $\vec a$ and $\vec b$ are in mutually opposite directions, then $\vec a\cdot\vec b =$ ____.

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.
• Negative → the vectors point in roughly opposite directions (angle greater than $90^\circ$). They oppose each other.

The Precise Statement

For two vectors $\vec{a}$ and $\vec{b}$ in any dimension:

where $\theta$ is the angle between them (between $0^\circ$ and $180^\circ$).

Since $|\vec{a}|$ and $|\vec{b}|$ are always positive lengths, the sign is entirely determined by $\cos\theta$:

Why This Matters in Exams

1. Check perpendicularity — If $\vec{a} \cdot \vec{b} = 0$, the vectors are orthogonal. A common exam shortcut.
2. Determine the angle type — Positive means acute, negative means obtuse, zero means right angle. …