How can line segments that don't meet be called "perpendicular"?

Question

Suppose there is a line segment from $(4,0)$ to $(6,0)$, and another line segment $(0,1)$ to $(0,2)$. They don't form an angle, so how are they "perpendicular"?

What actually is meaning of it? Like in triangles, they extend a side of obtuse angle to say the altitude is perpendicular to the side. So, I don't understand the criteria of perpendicular lines or line segments or rays. How come, if they are not even forming an angle, they are perpendicular?

I am new. I don't know coordinate geometry. I just used it a little to explain my problem without drawing.

Please explain simply and easily using figures rather than coordinate geometry.

Answer 1

You are working with an inadequate definition of angle, which requires the elements (segments, rays, or lines) in question to intersect before an angle is formed.

But a more general notion of angle between two suitable objects (which is useful when you come to deal with vectors) is as an operation (usually called a rotation) which when applied to one of the objects, makes the elements be the same with respect to some property. For example, dealing with rays or directed segments, the property with respect to which one might define an angle might be direction; that is, we may define an angle between two rays, say, as a rotation which when applied to either ray, makes both rays have the same orientation. There is an even more general notion of angle (where we give the rotation in question a direction, but that is not necessary for your question here).

Indeed, since your segments are not oriented, we may define an angle between two segments as a rotation which applied to either of the segments, makes them parallel. And we then say that two segments are parallel provided the lines they determine are parallel.

It then follows that two segments are perpendicular provided the lines they determine are perpendicular. For your question, each of the segments determines a line with equation $ax+by+c=0,$ where $a$ and $b$ do not simultaneously vanish, and $c$ can be any real number. We have three cases to consider. First, if $b\neq 0$ for both of these equations, then you can put them in the form $y=mx+c,$ where $m\neq 0$ and is called the slope of the line. These lines are perpendicular if and only if the product of their slopes is $-1.$ If $b=0$ in exactly one of the equations, then that equation represents a line vertical with respect to the rectangular coordinate system, so that they're perpendicular if and only if the second line is horizontal. This happens when $a=0.$ Finally, if in both equations $b=0,$ then the two of them are vertical, and therefore parallel.

Answer 2

Line segments define lines. If the lines are perpendicular, the segments are perpendicular, by definition. We have to assume that we are working in Euclidean (or pseudo-Euclidean) geometry. Otherwise ideas such as parallel and perpendicular become far more difficult.

Draw an arrow parallel to each line (segment) and transport it to the origin without changing its direction. If the arrows are perpendicular, then we can use that as a definition of perpendicularity.

Another way to think about this is to consider a plane lying perpendicular to one of our lines. If the other line lies parallel to the plane, it is perpendicular to the first line.