Vector and cartesian equations of curves (WACE Year 12 Mathematics Specialist Unit 3)

What this dot point is asking

SCSA wants you to move fluently between the vector form of a curve, its parametric component equations, and its cartesian equation, and to recognise the standard curves these describe.

From vector to parametric form

A curve in the plane can be written as a position vector that depends on a parameter:

As $t$ ranges over an interval, the tip of $\mathbf{r}(t)$ sweeps out the curve. The two component functions $x = x(t)$ and $y = y(t)$ are the parametric equations of the same curve.

Three-dimensional curves

The same idea extends to space, where a vector function $\mathbf{r}(t) = x(t)\,\mathbf{i} + y(t)\,\mathbf{j} + z(t)\,\mathbf{k}$ traces a curve in three dimensions. A single Cartesian equation can no longer capture such a curve; instead it is described by two relations between $x$, $y$ and $z$, or most naturally by keeping the parametric form. A helix, for instance, has $x = a\cos t$, $y = a\sin t$, $z = bt$: the first two components trace a circle while the third climbs steadily, so the path spirals up a cylinder of radius $a$. Parametric form is the only practical way to describe curves like this.

Eliminating the parameter

The resulting relation between $x$ and $y$, free of $t$, is the cartesian equation. Always note any restriction the parameter places on the range of $x$ or $y$, since the parametric curve may be only part of the full cartesian graph.

Recognising standard curves

Some parametrisations recur. A linear $\mathbf{r}(t) = \mathbf{a} + t\mathbf{b}$ traces a straight line. The pair $x = a\cos t$, $y = a\sin t$ traces a circle of radius $a$, since squaring and adding gives $x^2 + y^2 = a^2$. The pair $x = a\cos t$, $y = b\sin t$ traces an ellipse.

A translated circle uses $x = h + a\cos t$, $y = k + a\sin t$, which gives $(x - h)^2 + (y - k)^2 = a^2$, a circle of radius $a$ centred at $(h, k)$. A parabola can be parametrised as $x = at^2$, $y = 2at$, which eliminates to $y^2 = 4ax$. Knowing these standard forms lets you name a curve at a glance, which is often half the marks in a description question.

Direction of tracing and the parameter range

The vector form carries more information than the cartesian equation: it also records the direction and the part of the curve that is traced. As $t$ increases, the tip of $\mathbf{r}(t)$ moves along the curve in a definite sense; for $x = a\cos t$, $y = a\sin t$ the motion is anticlockwise starting from $(a, 0)$. A restricted range such as $0 \le t \le \pi$ traces only the upper half of the circle, so the cartesian equation must be accompanied by the inequality $y \ge 0$ to describe the same set of points. Always translate the parameter restriction into a restriction on $x$ or $y$.

Why both forms are useful

The cartesian equation is best for recognising the shape and for checking whether a point lies on the curve. The vector or parametric form is best for describing motion along the curve, for differentiating to get velocity (the next dot point), and for handling curves that are not functions, such as a full circle, where a single $y = f(x)$ rule cannot capture both halves. Fluency in moving between the two is a core SCSA expectation.