Dot Product and Orthogonality

Multiplying Vectors to Get a Number

The dot product takes two vectors and returns a single number (a scalar). For $u = (u_1, u_2)$ and $v = (v_1, v_2)$: $$u \cdot v = u_1 v_1 + u_2 v_2.$$ Multiply matching components and add. This simple operation encodes a surprising amount of geometry.

Its geometric meaning is the key: $u \cdot v = \|u\|\,\|v\|\cos\theta$, where $\theta$ is the angle between the vectors. So the dot product measures how aligned two vectors are:

• Positive — the vectors point in broadly the same direction ($\theta < 90°$).
• Zero — the vectors are orthogonal (perpendicular, $\theta = 90°$).
• Negative — they point in broadly opposite directions ($\theta > 90°$).

The orthogonality test is the most-used consequence: two nonzero vectors are perpendicular exactly when their dot product is zero. No angle computation needed — just check if $u \cdot v = 0$. This is how graphics code tests perpendicularity and how projections are computed.

The dot product also recovers length ($u \cdot u = \|u\|^2$) and lets you find the angle between vectors via $\cos\theta = \frac{u \cdot v}{\|u\|\,\|v\|}$. In machine learning, the closely related cosine similarity uses exactly this to measure how similar two feature vectors are — the backbone of search and recommendation.

Common pitfall: confusing the dot product (which returns a number) with scaling or componentwise multiplication that returns a vector, and 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 means the angle is acute or obtuse, never a right angle.