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Calculus II: Vector Calculus and Multivariable Analysis / Vectors and Curves in Space

Vectors in 2D and 3D: Operations, Length, and Direction

A portfolio of nn assets is a vector of weights w=⟨w1,w2,…,wn⟩\mathbf{w} = \langle w_1, w_2, \ldots, w_n \rangle summing to one; the portfolio return is the dot product of weights and asset returns. Before you can write down a covariance matrix, run gradient descent on a loss function, or derive the Black–Scholes Greeks, you need a fluent working knowledge of the objects that live in Rn\mathbb{R}^n. This note builds that foundation: what a vector is, how to add and scale vectors, how to measure length, and how to extract direction. Everything in Calculus II rests on this vocabulary.
Learning objectives. By the end of this note you should be able to:
  • define a vector in R2\mathbb{R}^2 and R3\mathbb{R}^3 in terms of its components and interpret it as a directed quantity;
  • perform vector addition and scalar multiplication both algebraically (componentwise) and geometrically (parallelogram rule, stretching);
  • compute the Euclidean norm ∥v∥\|\mathbf{v}\| and verify the triangle inequality ∥u+v∥≤∥u∥+∥v∥\|\mathbf{u} + \mathbf{v}\| \le \|\mathbf{u}\| + \|\mathbf{v}\|;
  • normalise a nonzero vector to obtain its unit vector and decompose any vector in terms of the standard basis i\mathbf{i}, j\mathbf{j}, k\mathbf{k};
  • extend all of the above to R3\mathbb{R}^3 using the 3D distance formula.

In this lesson

    1. The Vector Concept
    1. Vector Operations
    1. Length (Euclidean Norm)
    1. Unit Vectors and Standard Basis
    1. The Sphere and Ball in R3\mathbb{R}^3
    1. Vector Algebra: Key Properties Summary

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