First, you say that "t" and "r" are parameters but your equations have "s" and "r". There is no "t".
I will assume that you mean parameters "s" and "r".
You say you don't know how to convert them-where did you get this problem? It seems very strange that you would be give homework on a subject you have never seen!
Do you know what something like "\(\displaystyle A\vec{a}= \vec{b}\)" means Have you ever seen equation like these- which could also be written as \(\displaystyle x_1+ 0x_2+ x_3= s\), \(\displaystyle 3x_1+ x_2+ 3x_3= 0\), and \(\displaystyle x_1- x_2- rx_3= 0\)- before?
For the second form, we can interpret the matrix multiplication as the sum of dot products of the columns of the matrix with the components of the vector:
\(\displaystyle \begin{bmatrix}a & b & c \\ d & e & f\\ g & h & i\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}ax+ by+ cz\\ dx+ ey+ fz \\ gx+ hy+ iz \end{bmatrix}\)
can also be written as
\(\displaystyle \begin{bmatrix}a \\ d \\ g \end{bmatrix}x+ \begin{bmatrix}b \\ e \\ h \end{bmatrix}y+ \begin{bmatrix}c \\ f \\ i \end{bmatrix}z\).
Finally, the last form looks like the "augmented matrix" for solving a system of equations- you write the equation "ax+ by+ cz= u", "dx+ ey+ fz= v", and "gx+ hy+ iz= w" as the "augmented matrix"
\(\displaystyle \begin{bmatrix}a & b & c & u \\ d & e & f & v \\ g & h & i & w\end{bmatrix}\) (there should be a vertical line between the third and fourth columns but I don't know how to do that in Latex.)
and "row reduce" the matrix.
