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Below is my problem picture

Question image

Line $a$ runs parallel to $x$-axis.

Line $b$ runs parallel to $y$-axis.

When the two lines meet, then they become perpendicular to each other.

Now, slope of line $a$ is $$m_1=\tan 0 = 0$$

And slope of line $b$ is $$m_2=\tan 90 = 1/0$$

According to converse of perpendicularity of coordinate geometry :

Two lines will be perpendicular to each other if and only if the product of their slopes is equal to $-1$

That means $$m_1 \cdot m_2 = -1$$

But in my problem when we multiply $m_1$ and $m_2$ :

$$m_1 \cdot m_2 = 0 \cdot 1/0 =0$$

So the product becomes $0$, then please explain if I am missing something !!!!!

John Alexiou
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Rajat Jaiswal
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    $0\times \frac 10$ is undefined. The slope product formula doesn't make obvious sense when one line is vertical. – lulu Oct 14 '16 at 12:25
  • Vertical and horizontal lines are the "weird" cases in parallel lines. Normally, given two lines, they're parallel only if their slopes multiply to $-1$. But when a horizontal line and vertical line meet, they are always parallel. – Frank Oct 14 '16 at 12:38
  • @lulu Not "obvious" sense in an elementary way perhaps, but if you view that as the limiting case $\displaystyle \lim_{\theta \to \frac{\pi}{2}}\tan\theta \cdot \tan(\frac{\pi}{2} + \theta) = \lim_{\theta \to \frac{\pi}{2}}\tan\theta \cdot (-\cot \theta) = -1$, then it makes perfect sense. – Deepak Oct 14 '16 at 15:35

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Another way to frame whether two lines $L,M$ are perpendicular is to form two vectors pointing along each, then the lines are perpendicular iff the dot product of those two vectors is zero. [To form such a vector for a line $L,$ one takes two distinct points on $L$ and subtracts coordinatewise.]

For a line parallel to the $x$ axis one can use a vector like $(a,0)$ and for a line parallel to the $y$ axis a vector like $(0,b),$ where neither $a$ nor $b$ is zero since we subtracted different points each time. So here the dot product is $(a,0) \cdot (0,b)=a \cdot 0 + 0 \cdot b=0,$ and we see the lines are perpendicular by the dot product calculation.

One advantage of the dot product approach is that it works for any two lines, even if one or both are vertical (so no slope).

coffeemath
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