Question

in the first lecture of 18.02 why does x+2y+3z=0 produce a plane?

Answer 1

In a two dimensional plane, a set of points given will produce a line. In the three dimensional equation used there will be a plane created. When the professor draws objects on the graph with volume he is connecting all of the individual vectors together with their individual magnitude and direction. The professor does this in order to show how to calculate the magnitude of the vector he is creating in the three dimensions the vector travels in

Answer 2

One way to see this is to consider what you get when you keep one variable constant. Lets take z=k, where k is a constant.
\[x+2y=-3k\]This is an equation of a line. So all slices of this graph parallel to the x-y plane are straight lines. If we take slices parallel to the x-z plane, y=k=constant we get:\[x+3x=-2k\]So all slices of the graph parallel to the x-z plane are also straight lines. Similarly for slices parallel to the y-z plane. The only figure that gives straight lines in all level slices of it's graph is a plane.