# 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 ?

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In a right triangle with "horizontal" leg of length $1$, the "vertical" leg has length equal to the slope of the hypotenuse. Thus, you can easily "construct" $s_1$ and $s_2$ by simply attaching a horizontal-unit-leg right triangle to each line and looking at the vertical legs. Once the slopes are represented as segments, you can easily devise constructions of their sum, difference, product, and ratio. – Blue Mar 10 '12 at 12:31

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.

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An application is that in relativity, the combination of velocities $u$ and $v$ in one dimension gives velocity $(u+v)/(1+uv)$ (in units where $c=1$), rather than $u+v$ as in Galilean relativity. – Ben Crowell Mar 10 '12 at 14:19
Interesting. Feel free to edit my post and add anything if wish. – Inquest Mar 10 '12 at 14:34
Oops, I got that wrong. The relativistic addition of velocities is actually an application of the addition formula for the hyperbolic tangent, not the circular tangent. $\tanh(x+y)=(\tanh x+\tanh y)/(1+\tanh x \tanh y)$, where the sign in the denominator is opposite to the circular version. The quantity $\tanh^{-1}(v/c)$ is called the rapidity, and it adds linearly. – Ben Crowell Mar 11 '12 at 20:54