The aim of this question is to learn about the three-dimensional space $ R^3 $ and its subsets.
The three-dimensional space can be represented with the help of 3-coordinates in the cartesian system. Usually, these coordinates are x, y, and z-coordinates. The subsets of this three-dimensional space can be described with the help of constraint equations that restrict the domain or range of the space.
The subset region can have three possibilities. If all three coordinates are constrained and there is a definite unique solution to all of them, then the subset region represents a point. If two of them are constrained and the third one is open, then the subset region represents a plane. And if all of the axes have no unique solution under the given constraints, then the subset region is also a three-dimensional space.
The constraints that we use to find these subsets may be equations or inequalities. In the case of inequalities, we first find the constraint using the borderline equation, and then we apply the inequality condition to find the region of interest.
Recall the given equation:
\[ x \ = \ 10 \]
Now notice that $ R^3 $ is three-dimensional space and to describe a region in a three-dimensional space, we need to put constraints on all of the three cartesian coordinates. If we constraint only one of the coordinates and the other two are unconstrained (which is the case here), then the resulting region may be a plane.
In our case, the region represents a plain that spans the y and z coordinates from negative infinity to positive infinity. In short and simple words, the equation represents a yz-plane that cuts the x-axis at x = 10 mark.
The equation x = 10 represents a yz-plane in $ R^3 $ that cuts the x-axis at x = 10 mark.
Describe the region bound by the following equations in $ R^3 $ space.
\[ x^2 \ = \ 10 y \ … \ … \ … \ ( 1 ) \]
\[ y \ = \ 10 z \ … \ … \ … \ ( 2 ) \]
\[ z \ = \ 10 x \ … \ … \ … \ ( 3 ) \]
Substituting the value of z from equation (3) in equation (2):
\[ y \ = \ 10 (10x) \]
\[ \Rightarrow y \ = \ 100 x \ … \ … \ … \ ( 4 ) \]
Substituting the value of y from equation (4) in equation (1):
\[ x^2 \ = \ 10 ( 100x ) \]
\[ \Rightarrow x^2 \ = \ 1000 x \]
\[ \Rightarrow x \ = \ 1000 \]
Substituting this value in equation (3) and equation (4):
\[ y \ = \ 100 (1000) \]
\[ \Rightarrow y \ = \ \ 100000 \]
\[ z \ = \ 10 (1000) \]
\[ \Rightarrow z \ = \ 10000 \]
Hence we have a point:
( x, y, z ) = ( 1000, 100000, 10000 )
which required region represented by the above equations in $ R^3 $.