Coordinate of a Point in Three Dimension
Threedimensional space that can also be known as 3space or tridimensional space.
It is a geometric setting which contains three values are required to determine the position of an element .In physics and mathematics, a sequence of n numbers can be understood as a location in ndimensional space. When n = 3 it is called threedimensional Euclidean space. It is commonly represented by the symbol ℝ3. This acts as a threeparameter model of the physical universe in which all known matter exists. This space is only one example of a large spaces in three dimensions called 3manifolds. In this case, these three values are chosen from the terms width, height, depth, and length.
Points in 3 Dimension
On a two dimensional plane a point in the xyplane by an ordered pair that consists of two real numbers, an xcoordinate and ycoordinate, which denote signed distances along the xaxis and yaxis, respectively, from the origin, which is the point (0, 0). These axes, which are referred to as the coordinate axes, divided the plane into four quadrants. The properties of threedimensional space.

a point is represented by an ordered triple (x, y, and z) that consists of three numbers, an xcoordinate, a ycoordinate,

A zcoordinate in the twodimensional xyplane, these coordinates indicate the signed distance along the coordinate axes,

The xaxis, yaxis and zaxis, respectively, from the origin, denoted by o, which has coordinates (0, 0, and 0).
There is a onetoone correspondence between a point in xyzspace and a triple in R3, which is the set of all ordered triples of real numbers. This is known as a threedimensional rectangular coordinate system.
Example
The figure displays the point (2, 3, and 1) in xyzspace, denoted by the letter P, along with its projections onto the coordinate planes .The origin is denoted by the letter o.
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The point (2, 3, 1) in xyzspace, denoted by the letter P. The origin is denoted by the letter o. The projections of P onto the coordinate planes are indicated by the diamonds. The dashed lines are line segments perpendicular to the coordinate planes that connect P to its projections. Just as the xaxis and yaxis divide the xyplane into four quadrants, these three planes divide xyzspace into eight octants. Within each octant, all xcoordinates have the same sign, as do all ycoordinates, and all zcoordinates
How to Find Coordinates of a Point in a Three Dimensional Space
Finding a point in x,y,zspace can be difficult because, unlike graphing in the x,yplane, depth perception is required. The projection of a point (x, y, z) onto the x,yplane is obtained by connecting the point to the x,yplane by a line segment that is perpendicular to the plane, and computing the intersection of the line segment with the plane. Similarly, the projection of this point onto the xyplane is the point (0, y, z), and the projection of this point onto the xzplane is the point (x, 0, z). The figure shows these projections, and how they can be used to plot a point in x,y,zspace. One can first plot the point’s projections, which is similar to the task of plotting points in the x,yplane, and then use line segments originating from these projections and perpendicular to the coordinate planes to “locate” the point in x,y,zspace.
The Distance Formula Between the Two Points in Three Dimension
The distance between two points P1 = (x1, y1) and P2 = (x2, y2) in the xyplane is given by the distance formula,
d (P1, P2) = \[\sqrt{(x2 − x1)^{2} + (y2 − y1)^{2}}\]
Similarly, the distance between two points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) in xyzspace is given by the following generalization of the distance formula,
d (P1, P2) = \[\sqrt{(x2 − x1)^{2} + (y2 − y1)^{2} + (z2 − z1)^{2}}\]
This can be proved with the application of Pythagorean Theorem.
Solved Examples –
Question: Find the distance between P1 = (2, 3, 1) and P2 = (8, −5, 0)
Solution:
From the distance formula, we have.
d (P1, P2) =\[\sqrt{(8 − 2)^{2} + (5 − 3)^{2} + (0 − 1)^{2}}\]
= \[\sqrt{36 + 64 + 1}\]
= \[\sqrt{101}\] ≈ 10.05.
Question: Find the distance between the points (2,5, and 7) and (3, 4, 5).
Solution: d = \[\sqrt{(3 − 2)^{2} + (4(5))^{2} + (5 − 7)^{2}}\]
= \[\sqrt{1+81+4}\]
= \[\sqrt{86}\]
Q1. How do you Plot a Point in Three Dimensions and What are the Coordinates of a Point?
Ans. To plot the point (x, y, z) in threedimensions, we simply add the step of moving parallel to the zaxis by z units. That is, to plot a point (x, y, and z) in three dimensions, we follow these steps: Locate x on the xaxis. From that point the y axis moves y units
The exact location of a point is determined by coordinates on a twodimensional plane. The two axes of the coordinate plane are right angles to each other, called the x and y axis. The coordinates of a point shows how far it is from the axes.
Q2. What are Our 3 Dimensions and What Direction is XYZ?
Ans. Threedimensional space is a geometric threeparameter model in which there are three axes (x ,y and z axis)all known matter exists. These three dimensions are chosen from the terms length, width, height, depth, and breadth.
Zaxis is always the vertical axis and is positive upwards. the righthand rule where the thumb shows in the positive direction along the axis of rotation and the curled fingers indicate the positive rotation direction.