The value of $0\times\infty$ from Coordinate geometry

Question

The slope of $x$-axis is $0$ while that of $y$-axis is $\tan 90$,infinity. In coordinate geometry, the product of slope two perpendicular lines is $-1$. Does that apply to the coordinate axes as well, so is '$\infty\cdot 0 = -1$'?

Answer 1

You are right in the terms of first thing you said which is:

Product of Slope of two perpendicular lines is $-1$. Here slope is just the tangent of the anle given straight line make with positive $x-$axis.

Now, For the $x-$axis, the slope is $0$ as the angle between $x-$axis and $x-$axis is $0$ and $\tan 0=0$. For the same reason the slope of $y-$axis is $\tan 90=$ Not defined. People can have good arguments and discussion can go forever, but I shall avoid the term infinity here and use the term not defined.

Now whenever this term not defined$(\frac{0}{0})$ comes in mathematical context, it has only one purpose which is to inhibit us to proceed. When you use $(\frac{0}{0})$, you lose every right (From right I mean right to do arithmetic operation).

So, trying to compute $\frac{0}{0}\times 0$ is ultimately illegitimate in mathematics. Hence answer of your second Question is $NO$.

Answer 2

No. The statement "The product of the slopes of two perpendicular lines is $-1$" applies only to lines whose slope is neither $0$ nor $\infty$; "$0\cdot \infty$" is undefined. (Such weird entities arise, sort of, in calculus as indeterminate forms, but even then there isn't a single value of "$0\cdot\infty$".)

Answer 3

From just tan(0 degrees) and tan(90 degrees), horizontal and vertical, you can't just multiply 0 x $\infty$ and get 1.

Keep in mind $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

The product of a horizontal and vertical line would be $\tan(0)\tan(\frac{π}{2})$, where the angles are expressed in radians.

This is equal to $\frac{\sin(0)}{\cos(0)} * \frac{\sin\left(\frac{π}{2}\right)}{\cos\left(\frac{π}{2}\right)}$.

This is equal to $\frac{0}{1} * \frac{1}{0}$ which could arguably be equal to 1, but this is flawed.

To prove something like this, you would use a limit, in where you find $\lim_{x\to0}\tan(x)\tan\left(x+\frac{π}{2}\right)$.

However still, because it is an indeterminate form, the property of perpendicular lines' slopes having a product of 1 only applies to lines with real slopes.