Overview

This topic builds the prerequisite geometry for everything in Calculus II. It introduces vectors as mathematical objects that encode both magnitude and direction, extends the familiar calculus of a single variable to curves traced through $\mathbb{R}^2$ and $\mathbb{R}^3$, and develops the geometric vocabulary — dot products, cross products, arc length, curvature — that reappears in every later topic of this unit, from gradient descent in the optimisation topic to the divergence theorem in the integral theorems topic.

By the end of this topic you should be able to:

• perform vector algebra in $\mathbb{R}^2$ and $\mathbb{R}^3$ (addition, scalar multiplication, length, unit vectors) and connect length to the Euclidean distance formula;
• compute and interpret the dot product geometrically — angles, projections, and orthogonality — and recognise Pearson correlation as a normalised dot product;
• compute the cross product via the determinant formula, interpret its magnitude as an area and its direction via the right-hand rule, and evaluate the triple scalar product as a signed volume;
• write parametric and Cartesian equations of lines and planes in $\mathbb{R}^3$ and find distances between points, lines, and planes;
• differentiate and integrate vector-valued functions $\mathbf{r}(t)$, recover velocity and acceleration from a parametric curve, and compute arc length $L = \int_a^b \|\mathbf{r}'(t)\|\,dt$;
• compute the unit tangent $\mathbf{T}$, curvature $\kappa$, and osculating circle of a space curve.

1. Contents
2. The big picture