Question

Write the plane on parametric form:
2x-4y+z=0
What is the general algorithm for solving a "problem" of this sort?

Answer 1

Argh! This is a huge blockage of going further in my course! :(

Answer 2

Hmm. Just like any other thing. So, 3 variables, hence you need two parameters

let x=p, and let y=q

so, z= 4y-2z

hence, (x,y,z)=(p,q,4y-2z)

Get how it's done?

Answer 3

Not really :( I am aware that I need two parameters. Is there an unique solution for the equation on parametric form?

Answer 4

If so, the answer is supposed to be:
x=-s
y=t
z=2s+4t

Answer 5

I got no clue how they arrived at that though.

Answer 6

a plane can be defined like the following:

\[r = a + t_1b + t_2 c\]

we have 2x-4y+z=0

i want parameters that will give me a point where this is the case

so if we rearrange:

z = 4y - 2x

now choose parameters carefully so that this is the case:

let \[y = t_2 \text{ , } x = t_1\]
and using the earlier equation

\[z = 4t_2 - 2t_1\]

we now need \[r = a + t_1 b + t_2 c = \left(\begin{matrix}t_1 \\ t_2 \\4 t_2 - 2t_1\end{matrix}\right)\]
can you see that this is a possible solution?
\[a= \left(\begin{matrix}0 \\ 0 \\ 0\end{matrix}\right)\]

\[b = \left(\begin{matrix}1 \\ 0 \\-2\end{matrix}\right)\]
\[c = \left(\begin{matrix}1 \\ 0 \\ 4\end{matrix}\right)\]

Answer 7

how was that?

Answer 8

Extremely dedicated reply and easy to understand - thanks once again @eigenschmeigen !

Answer 9

;D its all in a days work for your local neighbourhood eigenschmeigen!