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Plane is an important topic in 3D Geometry. It is defined when the normal to it and distance to it from the origin are known or when the normal to it and the point lying on it are known, or when it passes through three points that are non-collinear in nature.
There are many points lying on the surface of the plane. When these points are connected, shapes can be formed, from a simple triangle to a more complex polygon with many numbers of sides.
The area is a feature of these shapes and their projection on a 3D coordinate plane gives rise to another shape.
3D coordinate system
The 3D coordinate space has three axes, namely x,y and z. These are all mutually perpendicular to each other.
Since there are three axes in a 3D coordinate system, there is a need for three coordinates to define a point in 3D. Once the coordinates are known, the point can be expressed in the form of (x,y,z), defining the position of that point with respect to the x,y and z axes, respectively.
Each of the three axes can be combined in pairs of two perpendicular axes to obtain a plane in 3D. This can be done by considering one of the axes to be zero. Therefore, considering x=0 gives the yz plane. The xz plane results when y=0. By taking z=0, the last plane, i.e., xy is obtained.
Line in 3D
A line in 3D coordinate space is a result of the joining of two or more points of the form (x,y,z).
When a line is drawn in 3D, it has both magnitude and direction. To define the direction, angles of inclination to the three axes are required to be known. It is inclined at specific angles to the axes as below,
x-axis: α
y-axis: β
z-axis: γ
The cosine of these angles to each of the three axes is represented as cosα, cosβ, cosγ. These are called directional cosines.
A line has two directions as it extends to either end. To get a unique set of direction cosines for a line, it is required to consider the line as a directed line, meaning that it has only one direction starting from the origin and extending in that one direction. When this is done, the direction cosines can be termed as l,m and n.
An important relation between the three direction cosines is that the sum of the squares of each of the direction cosines is equal to one. This can be mathematically written as l2+m2+n2=1.
Surface in 3D
Joining multiple lines together gives a surface in 3D. It can be inclined at any angle to three axes and have any number of sides.
Vector area is associated with a surface drawn in 3D. The vector area is the product of the normal area of the surface in 2D and the normal to that surface.
This is crucial for learning the projection of a plane area on 3 coordinate planes.
Projection in 3D
Projection means to take an original object and then project it onto another object along a projection line or plane.
In 3D, the projection can be done for a point, line, or plane.
Projection of a plane area on 3 coordinate planes can be brought about by first considering a surface and its area and then taking the corresponding projection of it on the 3 coordinate planes.
Projection of a plane area on 3 coordinate planes
To understand the process easily, first consider a simple surface in 3D to be a triangle. For this, draw three points in the 3D coordinate system and then join them using three lines. Let the area of this surface be A.
Now, take each of the axes and project this surface onto the three planes.
So, the projection of a plane area on 3 coordinate planes, xy, yz and zx, will result in three surfaces again.
Now, take the areas of these surfaces to be A1, A2 and A3 in each of the three planes.
The direction cosines to the normal to these surfaces are to be taken as cosα, cosβ, and cosγ, respectively.
The vector area of the surface will be the product of the normal area and the normal to surface. So, A1=Acosα, A2=Acosβ and A3=Acosγ.
Taking the square of each and adding,
A12+A22+A32=A2cos2+A2cos2+A2cos2
A12+A22+A32=A2[cos2α+cos2β+cos2]
Since cos2α+cos2β+cos2=1,
A12+A22+A32=A2
∴A2=A12+A22+A32
Conclusion:
Projection of a plane area on 3 coordinate planes is the sum of the squares of the areas of the surface in each of the planes in the 3D coordinate system. For a triangular surface, it is expressed in mathematical terms as A2=A12+A22+A32, where the term A is the area of surface and the terms A1, A2 and A3 are the areas of the projection of the plane surface on 3 coordinate planes. Apart from finding projection of the plane on 3 coordinate planes, the projection of the point on plane and line on plane can also be done and are more likely to be asked in the competitive exams.
