Essential Notes on Points in a 3D Space

Introduction

3D means three-dimensional. The term three-dimensional expresses the fact that there is a requirement of three values to find out the position of an element. The three Dimensions-Point in a 3D Space deals with the dimensions, namely length, width, and height. Some 3D shapes examples are cube, cylinder, sphere, cuboids, triangular prism, hexagonal prism, etc. It gives the illusion of depth and shows the pictorial representation on a two-dimensional medium when the illusion is elevated by stereoscopic means. In short, it is the solid figure also has depth and thickness and 3D space is a geometry setting with three values known as parameters.

3-Dimensional Geometry

The three-dimensional geometry involves the shapes of mathematics and is used to represent either point or a line; also, it can be a plane in a 3D space. It is somewhere similar to that of the two-dimensional coordinate geometry. The Three Dimensions-Point in a 3D Space involves the x-axis, y-axis, and z-axis. Across all three axes, these are mutually perpendicular with one another with the same units of length.

There is a requirement of the three parameters in a third space to find out the exact location of a point. Like the two-dimensional coordinate system, the origin is present, which is the point of intersection. The space is divided by these axes into eight octants. It is essential to know that the points are separated by a comma and are written under parentheses in a Cartesian coordinate system.

How can one represent a Point, Plane, and Line in 3-Dimensional Geometry?

As three-dimensional geometry represents a point, plane, and line, you will further read about it in brief that the representation works.

Representation of a point in 3D Geometry

In three-dimensional geometry, the point is represented either on a Cartesian form or a Vector form.

• Cartesian Form

In the Cartesian form in the three-dimensional geometry, the representation is done using three coordinates: x-axis, y-axis, and z-axis. So the coordinates in three-dimensional geometry are x, y, z. The point of x value is abscissa, the y value is known as ordinate, and the z value is known as applicate.

• Vector Form

The representation of a point P in the vector form is a position vector OP. It is written down in the form of,

OP=xi+yj+zk,

Where,

i, j, k are denoted as the unit vectors and the x-axis, y-axis and z-axis, respectively.

Representation of a line 

In a three-dimensional cartesian system, the equation is computed by two different methods. These two methods are given below.

This equation of a line that passes through the point ‘a’, which is parallel to a given vector ‘b’, is represented as  

r = a + λb

This equation of a line that passes through two given points named as ‘a’ and ‘b’ is shown as r = a + λ(b – a)

Representation of a plane 

In this representation, the equations are computed in three ways based on available inputs, which is about the plane.

This first plane equation is denoted as perpendicular distance d from the origin by having a normal unit vector as ˆn is r. n=d. This is known as a normal form.

The second equation is the equation of a plane that passes through three noncollinear points →

a,b→ and c is (r-a)[(b-a)*(c-a)]=0. This is an equation called through noncollinear lines.

This third equation is formed through the intersection of the two planes.

Essential concepts of 3 Dimensional Geometry

The essential concepts about three-dimensional geometry deal with direction ratio, distance formula, midpoint formula and section formula. Let us examine these individually.

Direction ratios

Point A, (a,b,c) is shown as a vector. This is with the position vector OA=ai+bj+ck. Here the direction ratios are a,b,c. This concerns the x-axis, y-axis, and z-axis as the ratios represent the vector line here.

Distance formula

It is the distance between the two points (x1,y1,z1 and x2,y2,z2). It is the shortest distance equal to the square root of the sum of the square of the difference, i.e. the x coordinates, y coordinates, and z coordinates of the given points.

Mid-point formula

To find the mid-point of the line of the points joining (x1,y1,z1) and (x2,y2,z2) is considered a new point. In this, the abscissa is the average of the values of x of the two given points. The ordinate is calculated as an average of the values of y of the given points. The midpoint is located between the two points, which lies on the line joining them.

Section formula

This formula is used to find the coordinates of a point that divides the line segment joining the point  (x1,y1,z1) and (x2,y2,z2). It is represented in the ratio m:n.

Conclusion

The concept of three dimensions-Point in a 3D Space helps understand the various operations performed on a point in the three-dimensional form. The formulas and concepts are necessary to understand the exact geometry settings in accordance with all three values of the x, y, and z axes. Plotting the points in the Cartesian plane correctly is necessary to gain the correct result on the exact point.