Find Find the point in this plane where it intersects the z axis

[montana111]

Homework Statement

find the point in this plane where it intersects the z axis (x=y=0).

P = {<x;y;z;> = <1;-1;0;> + a<1;-2;0;> + b<2;0;-1;> , for all a,b}

(by <l;m;n;> i mean to denote a column)

Homework Equations

The Attempt at a Solution

I talked to my TA last friday and he made it seem easy, but I don't remember what he said anymore. I have something that looks like

I set up a matrix that looks like

[<1;-1;0;> <1;-2;0;> <2;0;-1;>] [<1;a;b>] = [<0;0;2;>]

I'm not really sure what to do from here. It says it is asking for a point, does this mean which (a,b) works? What can you help me with?

Thanks in advance.

 

[lanedance]

find (a,b) where it works by row-reducing your matrix

then use (a,b) in
P= P(a,b)=<x,y,z>=<x(a,b),y(a,b),z(a,b)>

a & b are parameters in the parameterisation of the plane, each (a,b) corresponds to a point <x,y,z> on the plane P

 

[SammyS]

Homework Statement

find the point in this plane where it intersects the z axis (x=y=0).

P = {<x;y;z;> = <1;-1;0;> + a<1;-2;0;> + b<2;0;-1;> , for all a,b}

(by <l;m;n;> i mean to denote a column)
...

This is all you should need.

<x;y;z;> = <1;-1;0;> + a<1;-2;0;> + b<2;0;-1;>​

It says that

x = 1 + 1∙a + 2∙b

y = -1 - 2∙a + 0∙b

z = 0 + 0∙a - 1∙b
​

Now "Find the point in this plane where it intersects the z axis (x=y=0)." So, find what a & b make this happen?

Edited per ehild's note.

Thanks goes to ehild.

 

[ehild]

This is all you should need.

<x;y;z;> = <1;-1;0;> + a<1;-2;0;> + b<2;0;-1;>​

It says that

x = 1 + 1∙a + 2∙b

y = -1 - 2∙a - 2∙b

z = 0 + 0∙a - 1∙b
​

SammyS,

There is a typo in your equations (shown in red)

It should be

y=-1-2∙a +0∙b

ehild