We have already seen that a convenient way to describe a line in three dimensions is to provide a vector that "points to'' every point on the line as a parameter $t$ varies, like $$\langle 1,2,3\rangle+t\langle 1,-2,2\rangle =\langle 1+t,2-2t,3+2t\rangle.$$ Except that this gives a particularly simple geometric object, there is nothing special about the individual functions of $t$ that make up the coordinates of this vector—any vector with a parameter, like $\langle f(t),g(t),h(t)\rangle$, will describe some curve in three dimensions as $t$ varies through all possible values.

Example 15.1.1 Describe the curves $\langle \cos t,\sin t,0\rangle$, $\langle \cos t,\sin t,t\rangle$, and $\langle \cos t,\sin t,2t\rangle$.

As $t$ varies, the first two coordinates in all three functions trace out the points on the unit circle, starting with $(1,0)$ when $t=0$ and proceeding counter-clockwise around the circle as $t$ increases. In the first case, the $z$ coordinate is always 0, so this describes precisely the unit circle in the $x$-$y$ plane. In the second case, the $x$ and $y$ coordinates still describe a circle, but now the $z$ coordinate varies, so that the height of the curve matches the value of $t$. When $t=\pi$, for example, the resulting vector is $\langle -1,0,\pi\rangle$. A bit of thought should convince you that the result is a helix. In the third vector, the $z$ coordinate varies twice as fast as the parameter $t$, so we get a stretched out helix. Both are shown in figure 15.1.1. On the left is the first helix, shown for $t$ between 0 and $4\pi$; on the right is the second helix, shown for $t$ between 0 and $2\pi$. Both start and end at the same point, but the first helix takes two full "turns'' to get there, because its $z$ coordinate grows more slowly. $\square$