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so I went to a multivariable calculus lecture at MIT and the professor (who was barely speaking English lol), wrote that x+z=1 is a plane.

Now, can someone please explain that? I think I understand but years of high school algebra and indoctrination prevent me from thinking more freely.

By Averagemathgeek (Averagemathgeek) on Friday, September 17, 2004 - 11:15 pm: Edit

The professor is saying that the graph of x+z=1 in three-dimensions is plane.

One method of looking at this is to go from what is simple to what is not as simple. First start in the two dimensional space. Let this space look exactly like the standard xy-plane, except exchange the y-axis with the z-axis. If you graph x+z=1 on this plane, it will obviously be a straight line. Now you add the third dimension, y. First notice that if (x,z) satisfies x+z=1, then (x,y,z) will satisfy that equation in 3d because y does not affect the equality. So when you extend the equation to 3d, it will become a plane.

Hope this clears things up.

By Cherrybarry (Cherrybarry) on Friday, September 17, 2004 - 11:30 pm: Edit

so are there an infinite number of planes, since y can take on any value?

By Averagemathgeek (Averagemathgeek) on Friday, September 17, 2004 - 11:45 pm: Edit

No, there are an infinite number of lines. These come together to form a single plane (due to the nature of x+z=1).

Here is a way to visualize this. Take a sheet of paper. Let the perpendicular sides represent the x and z axis. Draw some line on the paper. This represents x+z=1 on the xz-plane. To incorporate the y-axis, you need to first realize where that axis is. The y-axis is orthogonal to its x- and z- counterparts. So, if the piece of paper (remember: this represents the xz-plane) is laying flat on a desk, the y-axis is pointing towards the sky. Move the paper up this y-axis. If you visualize the line moving up the line you will see it forms a plane. This is the graph of x+z=1 in xyz-space.

By Ay_Caramba (Ay_Caramba) on Friday, September 17, 2004 - 11:49 pm: Edit

edited

By Tongos (Tongos) on Saturday, September 18, 2004 - 12:08 am: Edit

it is a plane because no matter what the y the x function on z will always stay the same. so what you have is a bunch of the same functions (lines) stacked on top of one another in the y direction. atleast i think that is what it is.

By Cherrybarry (Cherrybarry) on Saturday, September 18, 2004 - 03:50 pm: Edit

so the plane is parallel to the y-axis? and y can take on any value without affecting the equation since the plane is parallel to its axis?

and if x=z, then the plane contains the y-axis?

By Tongos (Tongos) on Saturday, September 18, 2004 - 05:23 pm: Edit

well x+z=1, it will never intersect the y axis if that's what you mean.

x=z does contain the y axis in space (xyz) but doesnt contain y when you think of it in an xz plane, because y becomes non- existent.

By Tongos (Tongos) on Saturday, September 18, 2004 - 05:33 pm: Edit

its like saying x+0y=z