Hi students, in this chapter we will learn about introduction to 3 Dimensional Co-ordinate System. Till now you have studied about only 2D, here we will learn about 3D. In 3D we will have 3 axes, which are X axis, Y axis which you are comfortable with already. Here we will have extra one axis that is Z axis. And all these 3 axes are mutually perpendicular to each other. X axis, Y axis and Z axis. Here for everything we add Z co-ordinate extra. What is our origin in 2D? Origin in 2D is zero, zero. In 3D zero, zero, zero. Here we add one zero extra. Apart from lines and points, here we will have planes also. What do I mean by planes? Let us look at our first plane. So, the plane that contains X and Y axis is the XY plane. X and Y axis are two lines, the plane in which these two lines are there, this is called XY plane. Similarly the plane in which Y axis and Z axis are there, that is called YZ plane. Similarly, if we look at here this is our ZX plane, which consists of X axis and Z axis. Now, we have planes and their co-ordinated axes and our origin.
Let’s look at the next concept. Our next concept is marking a point a, b, c. So, in 2 dimension it is very easy if you have to mark 2, 3 or a, b something like that. In 3D if you have to mark a, b, c, how do we do it? We have to follow a procedure. What are the procedures? The first procedure is construct a cuboid. The second procedure I will tell you after doing the first procedure. Let’s look at how to construct a cuboid. So, we have to take our X axis, Y axis and Z axis. In this, this is our origin which is zero, zero, zero. So, we have to mark our point a, b, c. First I will take a, 0, 0 on axis. Then I will take 0, b, 0 on Y axis. Similarly I will take 0, 0, c on Z axis. Now, I have 3 points. Now, what will I do is from X axis and Y axis. I will draw two perpendiculars such that they will meet at a point. Let’s look at those two perpendiculars from these two that they are meeting at a, b, 0. From X axis and Y axis, if you draw two perpendiculars they are meeting at a, b, 0. Similarly we will draw two perpendiculars from X and Z axis. They will meet at a, 0, c. Now we have two more axes left. Those two if you draw perpendiculars they will meet at 0, b, c. Now, we have three different points, I told we have to construct a cuboid, if I have to complete the cuboid I have to draw three more lines which they will meet at this point, which is our required point a, b, c. So, what do we do? We take the appropriate dimension on X axis, Y axis and Z axis. We try to construct a cuboid. One end of the corner will be origin, other corner will be our point a, b, c. I hope this is simple. Now, next way we mark it is, move a units along X axis and b units parallel to Y axis and c units parallel to Z axis. I said, a units along X axis, b units parallel to Y axis and c units parallel to Z axis. You will reach the point a, b, c. I hope you understood, how to mark a point a, b, c.
Let’s look at the next thing, equations of the planes. So, for marking the equations of the planes. Let’s take all the points that are there on XY plane. If you take all the points on XY plane, we observe one simple property for all of them. I hope you will now observe it much better. All the Z co-ordinates on the XY plane are zero. So, we say the XY plane equation is Z equal to zero. Now, let’s look at the next plane if we look at YZ plane, all the X co-ordinates on the YZ plane will be zero. So, our equation of YZ plane is X is equal to zero. Similarly, if we look at the ZX plane, all the points on ZX plane will have Y co-ordinate zero. So, the equation of ZX plane is Y equal to zero. I suppose you are clear with all these equations.
Let’s look at some simple, simple formulae which are nothing but small extension or Z co-ordinate extra for 2D geometry. The first formula is distance formula. I hope this is our simple formula in 2D geometry square root of x2 minus x1 whole square plus y2 minus y1 whole square. We will simply add z2 minus z1 whole square to get our distance formula in 3D geometry. Similarly, our ratio formula, if we have m is to n ratio for two points, our ratio formula is this one, mx2 plus nx1 upon m plus n, my2 plus ny1 upon m plus n. Here we add one Z co-ordinate extra which is mz2 plus nz1 upon m plus n. The last one is the most easiest formula, our centroid formula. Centroid formula is sum of the co-ordinates by 3. In 2D, it is x1 plus x2 plus x3 by 3, y1 plus y2 plus y3 by 3. In 3D, we see, simply add the Z co-ordinates extra that is z1 plus z2 plus z3 by 3.
I hope you learned basics of 3D geometry and very simple formulae.
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