Dot Product and Orthogonality

Multiplying Vectors

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.

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

Orthogonality and Angles

Two nonzero vectors are orthogonal (perpendicular) exactly when $u \cdot v = 0$. No angle computation is needed.

The dot product recovers length via $u \cdot u = \|u\|^2$. You can find the angle between vectors using:

$\cos\theta = \frac{u \cdot v}{\|u\|\|v\|}$

In machine learning, cosine similarity uses this exact formula to measure how similar two feature vectors are for search.

Pitfall: The dot product returns a number, never a vector. Orthogonal always means a dot product of zero.