The Dot Product: Angles, Projections, and Correlation

The Pearson correlation between two return series is the dot product of their standardised versions. This is not a metaphor — it is the mathematical definition. Understanding the dot product geometrically as a measure of "how parallel" two vectors are translates directly into understanding why diversification reduces portfolio risk (uncorrelated assets have zero dot product), why high-correlation portfolios concentrate rather than hedge (near-parallel vectors), and why the covariance matrix entry $\Sigma_{ij}$ measures co-movement in the direction of joint returns. Everything in risk and portfolio theory that involves "correlation" is, at its core, a dot product.

• compute the dot product $\mathbf{v} \cdot \mathbf{w}$ algebraically (componentwise) and verify its basic algebraic properties (commutativity, distributivity, homogeneity);
• state and apply the geometric formula $\mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\|\|\mathbf{w}\|\cos\theta$ to find angles between vectors;
• characterise orthogonality ($\mathbf{v} \cdot \mathbf{w} = 0$) and parallelism ($|\mathbf{v} \cdot \mathbf{w}| = \|\mathbf{v}\|\|\mathbf{w}\|$) in terms of the dot product;
• state and use the Cauchy–Schwarz inequality $|\mathbf{v} \cdot \mathbf{w}| \le \|\mathbf{v}\|\|\mathbf{w}\|$;
• compute the projection of $\mathbf{v}$ onto $\mathbf{w}$ and decompose $\mathbf{v}$ into a component parallel to $\mathbf{w}$ and a component orthogonal to $\mathbf{w}$;
• explain Pearson correlation as a normalised dot product.

1. 1. The Dot Product: Definition and Algebraic Rules
2. 1. The Geometric Formula: Angles
3. 1. Orthogonality, Parallelism, and Cauchy–Schwarz
4. 1. Projection
5. 1. Direction Cosines
6. 1. Correlation as a Normalised Dot Product