Why should I care about the symmetric equation of a line?

Question

I'm currently TAing for a multivariable calculus course. When I think of a line, I think of the span of a single vector, potentially shifted away from the origin. In this way, the most natural representation of a line (to me) is its vector form:

$$P + tv$$

where I first move to a point $P$ on my line, and then move in the direction of some vector $v$. And if I'm in $\mathbb{R}^3$ where $P = \langle x_0,y_0,z_0\rangle$ and $v = \langle v_1, v_2,v_3\rangle$, I can understand writing the line in parametric form:

$$\begin{cases}x = x_0 + tv_1 \\ y = y_0 + tv_2 \\ z = z_0 + tv_3\end{cases}$$

because we're just being explicit about components. However, if each $v_i$ is nonzero, there's the symmetric equation of a line

$$\frac{x-x_0}{v_1} = \frac{y-y_0}{v_2} = \frac{z-z_0}{v_3}.$$

Why would I (or my students) ever use this format to describe a line? I see geometry in the vector equation, but I see no geometry in the symmetric equation. Why not just use vector equations or parametric equations?

Answer 1

Take for example the curve $\mathbf{r}(t) = (1-\tan^2t,3+4\tan^2t,-2+7\sec^2t)$ Many students might not know what a curve with crazy trig functions might look like, but this still represents a line segment. What the symmetric form teaches is that the exact parametric equation describing how fast one component goes is unimportant. What determines the shape is the $\textbf{relationship}$ between the variables, i.e. how fast one component is going relative to the other. In other words, what makes a line is not the $t\mathbf{v}$, it's the fact that each variable has a linear relationship with each of the others.

Answer 2

When one sees a system of equations of the form $$\cdots=\frac ab=\frac xy=\cdots,$$ a linear relationship of the sequence $\ldots a,x,\ldots$ and the sequence $\ldots b,y,\ldots$ should immediately jump to the foreground of the mind. The system simply says that although two variables may change, their ratio is always fixed. That says that the variables are directly proportional, or in other words linearly related.

The symmetric form of the equation of a line has this form. It tells us that the the ratio between an arbitrary displacement of a point along the coordinate axes and the speed in that direction is fixed for all time. This is just a generalisation of how we think of a line in one dimension. We require that the "projection" of the point onto each of the coordinate axes should move along their several lines uniformly, that is, always severally covering equal distances in equal times.

There could be nothing more descriptive of a line!

In general, do not despise an alternative presentation of an object, especially if it's made it into "standard" textbooks -- chances are it must have proved its worth to be able to make it that way. It's sometimes more convenient to work with one representation than the other, so one should regard them all with as near equal importance as possible.