Vectors & Dot Products

Vectors encode direction and magnitude; the dot product measures alignment and drives projections, correlation and factor exposures.

1. The idea

A vector $v \in R^{n}$ is an ordered list of numbers that can represent a position, a portfolio of weights, or a set of factor exposures. Its length $∥ v ∥ = v^{T} v$ measures magnitude, and direction is what remains after normalizing by that length.

The dot product $u \cdot v = \sum_{i} u_{i} v_{i} = ∥ u ∥∥ v ∥ cos θ$ links algebra to geometry: it is zero exactly when vectors are orthogonal, positive when they point in similar directions, and negative when they oppose each other.

In quant work, dot products appear constantly: portfolio return is a dot product of weights and asset returns, and correlation is a normalized dot product of centered return vectors.

2. Why it matters

Every covariance matrix, regression and PCA computation reduces to dot products, so fluency here is a prerequisite for every later topic in this section.

3. Core mathematics

Dot product

u \cdot v = i = 1 \sum n u_{i} v_{i}

bilinear, symmetric

Norm

∥ v ∥ = v \cdot v

Angle

cos θ = \frac{u \cdot v}{∥ u ∥∥ v ∥}

Cauchy-Schwarz

∣ u \cdot v ∣ \leq ∥ u ∥∥ v ∥

equality iff parallel

4. How to recognize it

Think of this when…

Portfolio return as a weighted sum of asset returns
Questions about correlation, cosine similarity or orthogonal signals
Any phrase like 'weighted sum' or 'linear combination'

5. Worked example

Worked example

Let $u = (1, 2, 2)$ and $v = (2, 0, 1)$ . Find $u \cdot v$ , $∥ u ∥$ , and the angle between them.

1
Compute the dot product.
$u \cdot v = 1 (2) + 2 (0) + 2 (1) = 4$
2
Compute norms.
$∥ u ∥ = 1 + 4 + 4 = 3, ∥ v ∥ = 4 + 0 + 1 = 5$
3
Solve for the angle.
$cos θ = \frac{4}{3 5} \approx 0.596 \Rightarrow θ \approx 53. 4^{\circ}$
Answer: u·v = 4, ‖u‖ = 3, θ ≈ 53.4°

6. Common mistakes

Confusing the dot product (a scalar) with elementwise multiplication (a vector)
Forgetting to normalize before interpreting a dot product as a correlation or cosine
Assuming orthogonal implies independent — true only jointly with distributional assumptions

7. Quick check

If $u \cdot v = 0$ for nonzero vectors, they are:

Cauchy-Schwarz says $∣ u \cdot v ∣$ is at most:

8. Practice

6 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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