Answer:
Hence, none of the options presented are valid. The plane is represented by
.
Step-by-step explanation:
The general equation in rectangular form for a 3-dimension plane is represented by:
![a\cdot x + b\cdot y + c\cdot z = d](https://tex.z-dn.net/?f=a%5Ccdot%20x%20%2B%20b%5Ccdot%20y%20%2B%20c%5Ccdot%20z%20%3D%20d)
Where:
,
,
- Orthogonal inputs.
,
,
,
- Plane constants.
The plane presented in the figure contains the following three points: (2, 0, 0), (0, 2, 0), (0, 0, 3)
For the determination of the resultant equation, three equations of line in three distinct planes orthogonal to each other. That is, expressions for the xy, yz and xz-planes with the resource of the general equation of the line:
xy-plane (2, 0, 0) and (0, 2, 0)
![y = m\cdot x + b](https://tex.z-dn.net/?f=y%20%3D%20m%5Ccdot%20x%20%2B%20b)
![m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
Where:
- Slope, dimensionless.
,
- Initial and final values for the independent variable, dimensionless.
,
- Initial and final values for the dependent variable, dimensionless.
- x-Intercept, dimensionless.
If
,
,
and
, then:
Slope
![m = \frac{2-0}{0-2}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B2-0%7D%7B0-2%7D)
![m = -1](https://tex.z-dn.net/?f=m%20%3D%20-1)
x-Intercept
![b = y_{1} - m\cdot x_{1}](https://tex.z-dn.net/?f=b%20%3D%20y_%7B1%7D%20-%20m%5Ccdot%20x_%7B1%7D)
![b = 0 -(-1)\cdot (2)](https://tex.z-dn.net/?f=b%20%3D%200%20-%28-1%29%5Ccdot%20%282%29)
![b = 2](https://tex.z-dn.net/?f=b%20%3D%202)
The equation of the line in the xy-plane is
or
, which is equivalent to
.
yz-plane (0, 2, 0) and (0, 0, 3)
![z = m\cdot y + b](https://tex.z-dn.net/?f=z%20%3D%20m%5Ccdot%20y%20%2B%20b)
![m = \frac{z_{2}-z_{1}}{y_{2}-y_{1}}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7Bz_%7B2%7D-z_%7B1%7D%7D%7By_%7B2%7D-y_%7B1%7D%7D)
Where:
- Slope, dimensionless.
,
- Initial and final values for the independent variable, dimensionless.
,
- Initial and final values for the dependent variable, dimensionless.
- y-Intercept, dimensionless.
If
,
,
and
, then:
Slope
![m = \frac{3-0}{0-2}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B3-0%7D%7B0-2%7D)
![m = -\frac{3}{2}](https://tex.z-dn.net/?f=m%20%3D%20-%5Cfrac%7B3%7D%7B2%7D)
y-Intercept
![b = z_{1} - m\cdot y_{1}](https://tex.z-dn.net/?f=b%20%3D%20z_%7B1%7D%20-%20m%5Ccdot%20y_%7B1%7D)
![b = 0 -\left(-\frac{3}{2} \right)\cdot (2)](https://tex.z-dn.net/?f=b%20%3D%200%20-%5Cleft%28-%5Cfrac%7B3%7D%7B2%7D%20%5Cright%29%5Ccdot%20%282%29)
![b = 3](https://tex.z-dn.net/?f=b%20%3D%203)
The equation of the line in the yz-plane is
or
.
xz-plane (2, 0, 0) and (0, 0, 3)
![z = m\cdot x + b](https://tex.z-dn.net/?f=z%20%3D%20m%5Ccdot%20x%20%2B%20b)
![m = \frac{z_{2}-z_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7Bz_%7B2%7D-z_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
Where:
- Slope, dimensionless.
,
- Initial and final values for the independent variable, dimensionless.
,
- Initial and final values for the dependent variable, dimensionless.
- z-Intercept, dimensionless.
If
,
,
and
, then:
Slope
![m = \frac{3-0}{0-2}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B3-0%7D%7B0-2%7D)
![m = -\frac{3}{2}](https://tex.z-dn.net/?f=m%20%3D%20-%5Cfrac%7B3%7D%7B2%7D)
x-Intercept
![b = z_{1} - m\cdot x_{1}](https://tex.z-dn.net/?f=b%20%3D%20z_%7B1%7D%20-%20m%5Ccdot%20x_%7B1%7D)
![b = 0 -\left(-\frac{3}{2} \right)\cdot (2)](https://tex.z-dn.net/?f=b%20%3D%200%20-%5Cleft%28-%5Cfrac%7B3%7D%7B2%7D%20%5Cright%29%5Ccdot%20%282%29)
![b = 3](https://tex.z-dn.net/?f=b%20%3D%203)
The equation of the line in the xz-plane is
or ![3\cdot x + 2\cdot z = 6](https://tex.z-dn.net/?f=3%5Ccdot%20x%20%2B%202%5Ccdot%20z%20%3D%206)
After comparing each equation of the line to the definition of the equation of the plane, the following coefficients are obtained:
,
,
, ![d = 6](https://tex.z-dn.net/?f=d%20%3D%206)
Hence, none of the options presented are valid. The plane is represented by
.