Through a given point on a line, there is one and only one line perpendicular to the given line.

The hints that it gave was "Do not limit your thinking to just two dimensions". If it just a regular line there will only be one perpendicular line right?

I don't get what you mean by holding up a pencil. I hold up one and I don't see anything?

hold a pencil up. A real pencil. I bet you can make perpendiculars in many directions, because the pencil is in a three dimensional space.

Let line segment m and n be drawn perpendicularly to point b on line l.

Then ,
Angle mbl =90degree
And angle nbl=90degree
So the two line segment parallel
But two parallel line have no common point
This is contradiction
Therefore,only one perpendicular line can be draw

Ah, the wonders of geometry! While it may seem like there's only one possible perpendicular line to a given line through a certain point, let me introduce you to the wild world of three dimensions!

Imagine this: You're standing on a flat road, and there's a line running across it. Now, if you draw a line straight up from a point on the road, there's only one way to create a perpendicular line to the road in two dimensions. However, if we venture into the third dimension, things get a bit more wild!

In our three-dimensional world, that line running across the road is now a plane stretching infinitely in all directions. And just like before, when you draw a line straight up from your point on the road, there's still one perpendicular line that intersects with the plane. But guess what? If we tilt that line just a tad bit to the side, it becomes a completely different perpendicular line!

So, in the magical realm of three dimensions, we can have an infinite number of perpendicular lines through a given point on a line. It's like a big circus of possibilities, defying the constraints of two-dimensional thinking. So, don't limit yourself to just two dimensions; let your imagination soar through the wonderful world of three dimensions, where perpendicular lines can be as abundant as clowns in a clown car!

To understand why there is only one line perpendicular to a given line through a given point, we need to consider the concept of perpendicularity in a higher dimension. The hint about not limiting your thinking to two dimensions is significant in this context.

In two-dimensional space, a line is defined by two points or by an equation of the form y = mx + b. If we have a given line, we can find the slope of that line. Let's say the slope of the given line is m1.

Now, let's consider a point P on the given line. To find a line perpendicular to the given line through point P, we need to find a line with a slope that is the negative reciprocal of m1. Let's call this slope m2.

In two dimensions, when you flip the sign of the slope and take its reciprocal, you get the negative reciprocal. For example, if the slope of the given line is 2/3, the slope of the line perpendicular to it will be -3/2.

However, the hint suggests that we should think beyond two dimensions. In three-dimensional space, a line can be defined by a point and a direction vector. The direction vector indicates the direction in which the line extends to infinity.

To find a line perpendicular to the given line through point P in three dimensions, we need to find a direction vector that is perpendicular to the direction vector of the given line. This can be done by taking the cross product of the direction vector of the given line with any other vector that is not parallel to it.

Since the cross product of two non-parallel vectors in three-dimensional space is unique, there will be one and only one direction vector perpendicular to the given line. This direction vector, along with point P, determines a unique line perpendicular to the given line through point P.

Thus, in any dimension, including three-dimensional space, there is only one line that is perpendicular to a given line through a given point.