Geometric meaning of the sum/product of slopes Consider two lines in the plane $\mathbb{R}^2$ of slopes $s_1$ and $s_2$.
Is there a geometric meaning of the sum $s_1+s_2$, difference $s_1-s_2$, product $s_1s_2$ or quotient $s_1/s_2$ of the slopes ?
Put differently, is there a geometric construction from the lines that makes these numbers arise naturally ?
 A: If you consider the two lines with slope $s_1$ and $s_2$, they are basically two lines with inclination $\theta_1$ and $\theta_2$ where $s = \tan(\theta)$. 
If you are looking at physical interpretation of the sum,product or quotient of the slopes, it is limited but here are a few salient points:
If you have 2 lines with slopes $s_1$ and $s_2$, the angle between the lines is given by $$\tan(\theta) = |\frac{s_1-s_2}{1+s_1s_2}|$$
Its not difficult to see why once you know that the slope is actually the tangent of the inclination with X axis.
If line $L_1$ makes angle $\theta_1$ with the X axis and  $L_2$ makes angle $\theta_2$ with the X axis, the difference between the 2 angles is $|\theta_1 - \theta_2|$ which can be written in terms of slope using $\arctan(s_1)-\arctan(s_2)$ which using the identity $\arctan(A) + \arctan(B) = \arctan(\frac{A+B}{1-AB})$ gives us the required answer.

Now, if you have 2 perpendicular lines, the angle between them is $\dfrac{\pi}{2}$. This can be written as $\tan(\dfrac{\pi}{2}) = |\dfrac{s_1-s_2}{1+s_1s_2}|$. This is only possible when the denominator tends to $\infty$ with the numerator tending to some number in $\mathbb{R}$. This gives us the condition (in terms of slope) for 2 lines to be perpendicular viz. $s_1s_2 = -1$

Its not hard to see that two lines having the same slope are parallel (and vice versa).
I might think of more such points and i'll fill them in if I remember.
