For $\lambda,\mu \in \mathbb R$ we define:
$\begin{align} \begin{pmatrix} 1\\ 1\\ 1\\ \end{pmatrix}+\lambda\begin{pmatrix} 1\\ 0\\ -4\\ \end{pmatrix}+\mu\begin{pmatrix} -1\\ -5\\ 0\\ \end{pmatrix} \end{align} \qquad (1)\quad $
and
$\begin{align} \begin{pmatrix} 1\\ 1\\ 1\\ \end{pmatrix}+\lambda\begin{pmatrix} 1\\ 0\\ -4\\ \end{pmatrix}+\mu\begin{pmatrix} -2\\ -5\\ 4\\ \end{pmatrix} \end{align} \qquad (2)\quad$
I want to check if $(1),(2)$ describe the same plane. To this end, it suffices to calculate the cartesian equation for each parametric equation and see whether they coincide or not.
For $(1)$: $\begin{align} \begin{pmatrix} 1\\ 1\\ 1\\ \end{pmatrix}+\lambda\begin{pmatrix} 1\\ 0\\ -4\\ \end{pmatrix}+\mu\begin{pmatrix} -1\\ -5\\ 0\\ \end{pmatrix}=\begin{pmatrix} x_1\\ x_2\\ x_3\\ \end{pmatrix} \end{align} $ which yields the system
$\begin{alignat}{3} 1&+\lambda&&-\mu &&&=x_1\\ 1&\;&&-5\mu &&&=x_2\\ 1&-4\lambda &&\;&&&=x_3 \end{alignat}$
Multiplying with the appropriate coefficients and taking the sum, I find that: $20x_1-4x_2+5x_3=21$
In the exact same way I obtain for $(2)$ that $20x_1-4x_2+5x_3=21$ and hence I conclude that the two equations describe the same plane.
Is this conclusion correct or did I miss something? Is there any other way to check whether two parametric equations describe the same plane?
Many thanks in advance for your time!
As an alternative method, given that the fixed points are the same, you could calculate the cross-product of the direction vectors for each plane and see if they are parallel. Indeed they are.