91Ó°ÊÓ

The equations of transformations are,

$$\begin{gathered} x={\mathrm{Î\pm }}_{1}u+{\mathrm{β}}_{1}v+{\mathrm{γ}}_{1}w \\ y={\mathrm{Î\pm }}_{2}u+{\mathrm{β}}_{2}v+{\mathrm{γ}}_{2}w \\ z={\mathrm{Î\pm }}_{3}u+{\mathrm{β}}_{3}v+{\mathrm{γ}}_{3}w \end{gathered}$$

The objective is to determine the transformation of a line$ax+by+cz=d$in xyz- coordinate system to uvw- system.

The Jacobian of a transformation using partial derivatives is computed as,

$$\begin{gathered} \frac{∂(x,y,z)}{∂(u,v,w)}=\left|\begin{array}{ccc}\frac{∂x}{∂u}& \frac{∂y}{∂u}& \frac{∂z}{∂u}\\ \frac{∂x}{∂v}& \frac{∂y}{∂v}& \frac{∂z}{∂v}\\ \frac{∂x}{∂w}& \frac{∂y}{∂w}& \frac{∂z}{∂w}\end{array}\right| \\ \frac{∂(x,y,z)}{∂(u,v,w)}=\left|\begin{array}{ccc}{\mathrm{Î\pm }}_1& {\mathrm{Î\pm }}_2& {\mathrm{Î\pm }}_3\\ {\mathrm{β}}_1& {\mathrm{β}}_2& {\mathrm{β}}_3\\ {\mathrm{γ}}_1& {\mathrm{γ}}_2& {\mathrm{γ}}_3\end{array}\right| \end{gathered}$$

To transform the equation of plane in uvw- system , substitute the equations of transformations in the equation of line.

$$\begin{gathered} ax+by+cz=0 \\ (a{\mathrm{Î\pm }}_1+b{\mathrm{Î\pm }}_2+c{\mathrm{Î\pm }}_3)u+(a{\mathrm{β}}_1+b{\mathrm{β}}_2+c{\mathrm{β}}_3)v+(a{\mathrm{γ}}_1+b{\mathrm{γ}}_2+c{\mathrm{γ}}_3)w=d \end{gathered}$$

For the purpose of proving the statement, let us assume the converse.

Let us consider that both coefficients are zero.

Write the system of equations in matrix form.

$\left|\begin{array}{ccc}{\mathrm{Î\pm }}_1& {\mathrm{Î\pm }}_2& {\mathrm{Î\pm }}_3\\ {\mathrm{β}}_1& {\mathrm{β}}_2& {\mathrm{β}}_3\\ {\mathrm{γ}}_1& {\mathrm{γ}}_2& {\mathrm{γ}}_3\end{array}\right|\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

The determinant of the matrix is the Jacobian of the transformation.

Hence, the system is consistent if Jacobian is non-zero.

Since the system has only trivial solution. This is not an accepted solution for a given plane.

Hence, the system has not any real solution, if the Jacobian is non-zero.

This means the system of equations does not exist.

Thus, our assumption is proved to be incorrect.

Therefore, the coefficients of the plane in uvw- system are non zero, and hence represent a plane, if Jacobian is non-zero.