equation of a line for a street lamp post

Question

I am trying to apply mathematics in my everyday life and I feel like starting with simple equations of a line and my case study for that will be street lamp posts.

So, if I try to apply equation of a straight line to a street lamp post as :

    y = mx + c, where m = 0 

Therefore, y = c because there is no slope for that lamp post.
Is it correct? Please guide me as I am trying to apply mathematics to everyday life.

Answer 1

$y(x) = 0x + c$ gives you a constant function that is c for all values of x. If you want an equation for a vertical line, well... it's no longer a function, because $f(x)$ is not unique and can take on multiple (see, infinite) values.

If you wanted to approach a vertical line, you'd see the slope factor would approach infinity as it becomes closer and closer to vertical.

Answer 2

In the Cartesian coordinate system, the equation of what you're trying to model (an object standing vertically) could be described with the following expression:

$$x(y)=C.$$

Here $x$ is a constant function of $y$. So, you think of $x$ is the dependent variable and $y$ as the independent variable and you are still looking at your coordinate system with the x-axis as your horizontal axis and the y-axis as your vertical axis. That might work.

If you want your lamp post to have a certain height, then restrict the domain of your function:

$$x(y)=C,\ 0\le y\le h$$

where $h$ is the height of your lamp post.

P.S.: In Cartesian coordinates, the slope of a vertical line is considered undefined.

Answer 3

A slope of $0$ gets you a horizontal line, not a vertical one. The “general” equation $y=mx+b$ isn’t quite so general as advertised since no vertical line can be represented by an equation of this form: vertical lines have equations of the form $x=c$, but there’s no combination of $m$ and $b$ values that will eliminate $y$ from the equation $y=mx+b$.

Depending on what you’re doing with these equations, you might choose to treat vertical lines as exceptions (at the risk of forgetting to consider them at all!) or start instead with the more general equation $Ax+By+C=0$. If $B\ne0$, this can obviously be converted to slope-intercept form, while if $B=0$, the line is vertical.

Geometrically, the slope $m$ is the tangent of the angle $\theta$ that the line makes with the positive $x$-axis. If you divide both sides of the equation $Ax+By+C=0$ by $\sqrt{A^2+B^2}$, so that in the resulting equation $A'x+B'x+C'=0$ we have $A'^2+B'^2=1$, these coefficients can be interpreted as the sine and cosine of this same angle.