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If two lines are perpendicular, so their slope multiplication is negative. And one of these has slope zero and another is infinity. My question is that the multiplication of infinity to zero can be negative? like sum of no. from 1 to infinity is negative?

Steven Alexis Gregory
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Digvijay Negi
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  • As you are considering the gradient, think of it as $\dfrac {\text {rise}}{\text {run}}$. We can rise as much as we want for run $0$ for the line $x=0$. So if we take the gradient to be $-\frac 10$, we have $-\frac 10 \cdot 0=-\cfrac 1{\not {0}} \cdot \not 0=-1$. Not the most satisfactory result but the only one I believe makes a bit of sense. – For the love of maths Dec 14 '18 at 06:33
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    That’s just flat-out incorrect. One tangent is horizontal and the other vertical, so the angle of intersection is $90°$. – KM101 Dec 14 '18 at 06:46

2 Answers2

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The fact that the product of the slopes of two perpendicular lines is $-1$ only applies when both slopes exists. A vertical line has no slope - its slope does not exists. Saying that the slope of a vertical line is infinity and concluding that $\infty \cdot 0 = -1$ requires a severly restricted view of the definition of infinity which does not apply to and does not contradict other situations.

One way around this would be to use tangent vectors instead of slopes. You can parameterize the parabola $x^2=4y$ as $F(t) = (t, \frac 14t^2)$. Then the tangent vector at $F(0)$ is $\vec u = (1,0)$. You can parameterize the parabola $y^2=4x$ as $G(t) = (\frac 14 t^2, t).$ Then the tangent vector at $G(0)$ is $\vec v = (0, 1)$. Since $\vec u \circ \vec v = 0$, it follows that the two tangents are perpendicular.

Steven Alexis Gregory
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The first curve is a parabola, tangent to x-axis at origin. The second is also the same parabola rotated around origin through $-90^{\circ} $ and is symmetric with respect to x-axis. Recognizing that the first is an inverse function of the second, there is symmetry of these two curves with respect to line $ x = y. $

So the angle between vertical/horizontal tangents at origin is $90^{\circ} $

Narasimham
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