How to Find the Coordinates of a Point in Three Dimensions
Three-dimensional space that can also be known as 3-space or tri-dimensional 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 n-dimensional space. When n = 3 it is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3. This acts as a three-parameter 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 3-manifolds. 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 xy-plane by an ordered pair that consists of two real numbers, an x-coordinate and y-coordinate, which denote signed distances along the x-axis and y-axis, 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 three-dimensional space.
a point is represented by an ordered triple (x, y, and z) that consists of three numbers, an x-coordinate, a y-coordinate,
A z-coordinate in the two-dimensional xy-plane, these coordinates indicate the signed distance along the coordinate axes,
The x-axis, y-axis and z-axis, respectively, from the origin, denoted by o, which has coordinates (0, 0, and 0).
There is a one-to-one correspondence between a point in xyz-space and a triple in R3, which is the set of all ordered triples of real numbers. This is known as a three-dimensional rectangular coordinate system.
Example
The figure displays the point (2, 3, and 1) in xyz-space, 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 xyz-space, 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 x-axis and y-axis divide the xy-plane into four quadrants, these three planes divide xyz-space into eight octants. Within each octant, all x-coordinates have the same sign, as do all y-coordinates, and all z-coordinates
How to Find Coordinates of a Point in a Three Dimensional Space
Finding a point in x,y,z-space can be difficult because, unlike graphing in the x,y-plane, depth perception is required. The projection of a point (x, y, z) onto the x,y-plane is obtained by connecting the point to the x,y-plane 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 xy-plane is the point (0, y, z), and the projection of this point onto the xz-plane is the point (x, 0, z). The figure shows these projections, and how they can be used to plot a point in x,y,z-space. One can first plot the point’s projections, which is similar to the task of plotting points in the x,y-plane, and then use line segments originating from these projections and perpendicular to the coordinate planes to “locate” the point in x,y,z-space.
The Distance Formula Between the Two Points in Three Dimension
The distance between two points P1 = (x1, y1) and P2 = (x2, y2) in the xy-plane 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 xyz-space 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 = \[\sqrt{(3 − 2)^{2} + (4-(-5))^{2} + (5 − 7)^{2}}\]
= \[\sqrt{1+81+4}\]
= \[\sqrt{86}\]
FAQs on 3D Coordinates of a Point: Complete Guide
1. What are the coordinates of a point in a three-dimensional space?
The coordinates of a point in a three-dimensional (3D) space are a set of three values, represented as an ordered triple (x, y, z), that specify the exact location of the point. These values represent the signed perpendicular distances of the point from the three mutually perpendicular coordinate planes: the YZ-plane, the XZ-plane, and the XY-plane, respectively.
2. How do you plot a point with given 3D coordinates like P(a, b, c)?
To plot a point P(a, b, c) in the 3D coordinate system, you follow these steps from the origin (0, 0, 0):
Move 'a' units along the x-axis.
From that position, move 'b' units in a direction parallel to the y-axis.
Finally, from this new position, move 'c' units in a direction parallel to the z-axis. The final point is the location of P(a, b, c).
3. What is the main difference between coordinates in a 2D plane and a 3D space?
The main difference lies in the number of axes and the dimensions represented. A 2D plane uses two perpendicular axes (x and y) to define a point's location with an ordered pair (x, y). In contrast, 3D space uses three mutually perpendicular axes (x, y, and z) to define a point's location with an ordered triple (x, y, z), adding the dimension of depth or height.
4. What are the coordinate planes in 3D geometry and how are they defined?
In 3D geometry, the three coordinate planes are flat surfaces that are formed by pairing up the three axes. They divide the space into eight regions called octants. The planes are:
The XY-plane: The plane containing the x-axis and y-axis. For any point on this plane, the z-coordinate is always zero (z = 0).
The YZ-plane: The plane containing the y-axis and z-axis. For any point on this plane, the x-coordinate is always zero (x = 0).
The XZ-plane: The plane containing the x-axis and z-axis. For any point on this plane, the y-coordinate is always zero (y = 0).
5. Why are 3D coordinates written as an 'ordered' triple (x, y, z)?
The term 'ordered' is crucial because the sequence of the coordinates matters. Each position in the triple (x, y, z) corresponds to a specific axis. Changing the order of the values changes the location of the point in space. For example, the point (2, 4, 5) is completely different from the point (5, 2, 4) because the distances measured along the x, y, and z axes are different in each case.
6. How do the signs (+ or -) of the coordinates determine a point's location in 3D space?
The signs of the x, y, and z coordinates determine which of the eight octants the point lies in. The origin (0, 0, 0) is the intersection of the three coordinate planes. For instance:
A point with all positive coordinates like (+, +, +) lies in the first octant.
A point with coordinates like (-, +, +) lies in the second octant.
A point on an axis will have two coordinates as zero, like (x, 0, 0) on the x-axis. A point on a plane will have one coordinate as zero, like (x, y, 0) on the XY-plane.
7. What is the significance of the 'right-hand rule' in a 3D coordinate system?
The right-hand rule is a convention used to define the orientation of the axes in a 3D system, ensuring consistency. If you curl the fingers of your right hand from the positive x-axis towards the positive y-axis, your thumb will point in the direction of the positive z-axis. This standard orientation is essential in Maths and Physics for correctly calculating vector products, torque, and magnetic fields, ensuring that formulas work universally.
