Vectors and coordinate geometry

Builds on

Required prerequisites, all the way down: 19

You will learn to

• Add vectors, multiply them by scalars and interpret both geometrically and in coordinates.
• Use the dot product to compute lengths, angles and projections, and to test perpendicularity.
• Write vector and parametric equations of lines in the plane and in space, and of planes in space.
• Prove geometric facts (such as that the diagonals of a parallelogram bisect each other) using vectors.

Definition

• A vector in ℝ² or ℝ³ is an ordered list of real numbers, pictured as an arrow that can be moved without changing its length or direction.
• The dot product of u = (u₁, …, uₙ) and v = (v₁, …, vₙ) is u · v = u₁v₁ + ⋯ + uₙvₙ.
• The length of v is |v| = √(v · v).
• The line through a point p with direction d ≠ 0 is the set {p + t d : t ∈ ℝ}.

Theorem

• u · v = |u| |v| cos θ, where θ is the angle between u and v.
• Cauchy–Schwarz inequality: |u · v| ≤ |u| |v|, and the triangle inequality |u + v| ≤ |u| + |v|.
• Non-zero vectors u and v are perpendicular exactly when u · v = 0.

Common misconceptions

• Thinking a vector has a fixed starting point; a vector records only magnitude and direction.
• Believing the dot product of two vectors is a vector; it is a number.
• Assuming |u + v| = |u| + |v| in general; equality holds only when one vector is a non-negative multiple of the other.

Leads to

Optional discoveries

• Linear algebra The rules for adding vectors and multiplying them by numbers are exactly the axioms of a vector space, which linear algebra studies in any dimension.
• Differential forms and Stokes' theorem The dot product measures work along a path and the cross product measures oriented area; differential forms generalise both to any dimension.

Learning pathways

• Geometry and topology

Further reading

• Tom M. Apostol, Calculus, Volume 1 (1967)