Dot Product and Angles

Multiplying Vectors into a Scalar

The dot product (scalar product) combines two vectors into a single number. For $u = (u_1, u_2)$ and $v = (v_1, v_2)$, $$u \cdot v = u_1 v_1 + u_2 v_2$$ (and analogously in higher dimensions: multiply matching components and sum). Despite its simplicity, it encodes the angle between vectors.

Its geometric identity is the key: $$u \cdot v = \|u\|\,\|v\|\cos\theta,$$ where $\theta$ is the angle between the vectors. So the sign of the dot product reveals the angle:

• Positive — acute angle ($\theta < 90°$), the vectors broadly agree in direction.
• Zero — right angle; the vectors are orthogonal (perpendicular).
• Negative — obtuse angle ($\theta > 90°$), the vectors broadly oppose.

The orthogonality test is the workhorse: two nonzero vectors are perpendicular exactly when their dot product is zero — no trigonometry required. Rearranging the identity gives the angle itself: $\cos\theta = \frac{u \cdot v}{\|u\|\,\|v\|}$. The dot product also recovers length, since $u \cdot u = \|u\|^2$.

The dot product underlies projection (the component of one vector along another is $\frac{u \cdot v}{\|v\|}$) and the Cauchy–Schwarz inequality $|u \cdot v| \le \|u\|\,\|v\|$, a cornerstone of geometry and analysis. Wherever "how aligned are these?" is asked, the dot product answers.

Common pitfall: thinking the dot product yields a vector, or forgetting that orthogonal means dot product zero. $u \cdot v$ is a single scalar $u_1 v_1 + u_2 v_2$, not a vector. And perpendicular vectors have $u \cdot v = 0$: a positive or negative dot product signals an acute or obtuse angle, never a right angle.

Two vectors from the origin with the angle theta marked; the accent dot-product value shrinks to zero as the angle opens to 90 degrees — orthogonality at u · v = 0.

$$u \cdot v = \|u\|\,\|v\|\cos\theta$$