Why the intercept form and normal form is not applicable in 3d space?

Question

Today one of my folks told me during studying 3d geometry that the intercept form and normal form are not applicable in the 3d space where in 2d they both hold good 1.intercept form: x/a + y/b =1 2. Normal form : xcos(a)+ ysin(a) =p

Answer 1

There are many ways to show that a single linear equation can’t represent a line in $\mathbb R^n$ for $n\gt2$. The Rank-Nullity Theorem, for instance, tells us that the nullity of the matrix $\small{\begin{bmatrix}a_1&\cdots&a_n\end{bmatrix}}$ is at least $n-1$, so the solution set of the equation $a_1x_1+\cdots+a_nx_n=b$, if it has any solution at all, can only be a line when $n=2$ or $n=1$, that is, when the entire space that you’re working in is a line or plane. (In higher-dimensional spaces, the generalizations of these equations describe hyperplanes.) However, we can also see that this is so by looking at something more basic: the degrees of freedom of a line, which is essentially the minimum number of values that you have to provide to specify a line uniquely.

A line in the plane has two degrees of freedom. Note how both of the forms of equation in your question have two parameters, both of which are essential to identifying the line. We can generalize the normal form to the form $ax+by=c$, since multiplying both sides of an equation by the same nonzero number doesn’t change its solution set. It might look like this has three parameters, but because of this multiplicative property, they’re not independent, and the underlying unique parameters are the two independent ratios between the coefficients of the equation, which we can write in the compact form $a:b:c$.

The parameters of the intercept form have a geometric interpretation as the intersections of the line with the $x$- and $y$-axes (hence its name). We can do something similar in three dimensions to count the degrees of freedom there. Consider a pair of perpendicular planes, say, the $x$-$y$ plane and $x$-$z$ plane. You should be able to see that the points at which a line intersects these two planes determine it uniquely.† Each of these points has two degrees of freedom—it’s an arbitrary point on a plane—so a line in three-dimensional space has two more degrees of freedom than one in 2-D. Unfortunately, going to three dimensions only adds one variable to the equation, so there are more parameters for the line than there are slots for them in the equation. To put it a bit differently any equation equivalent to $ax+by+cz=d$ has three degrees of freedom, but we need four to be able to describe a line.

As you add more dimensions, you get farther and farther behind. Each dimension adds two degrees of freedom to lines, but only one to linear equations. Generally speaking, in $n$ dimensions you need $n-1$ linear equations to specify a line. Now, you might encounter the symmetric form $${x-x_0\over a}={y-y_0\over b}={z-z_0\over c}.$$ Some people call this an equation, but strictly speaking it’s a compact way of writing a system of two equations, just as we wrote a pair of ratios in a more compact form above.

We can also give a quick geometric argument for why a single equation equivalent to $ax+by+cz=d$ can’t represent a line in 3-D. If we rewrite it as $(a,b,c)\cdot(x,y,z)=d$, we can interpret the equation as saying that its solution set consists of all points suc that their dot product with $\mathbf n=(a,b,c)$ is equal to $d$. Suppose that you have some point $\mathbf p$ other than the origin that satisfies this equation. Arbitrarily rotating this point about the axis defined by the direction $\mathbf n$ doesn’t change the value of their dot product, which depends only on their length and the angle between them. As you rotate this point, it traces out a circle centered on this axis. All points on this circle are solutions to the equation, so the equation cannot represent a straight line. In higher dimensions, there are even more ways to rotate $\mathbf p$ without changing its dot product with $\mathbf n$.

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† Lines that are parallel to either of the planes or have a common intersection with them both are an additional complication that I’m ignoring, just as I ignored the fact that lines parallel to the 2-D coordinate axes or ones that pass through the origin don’t have a proper intercept-form equation. It’s possible to accommodate them in this framework, but the extra complexity doesn’t add anything essential to the underlying idea.