# How to think of a plane

• Cankur
In summary, The plane y=z is one of three planes that can be formed between the three different axes and is represented by a line at 45 degrees in the y-z plane. It consists of all vectors of the form x(1,0,0) + y(0,1,1) and can intersect the x-axis and the line y=z at different angles depending on the perspective. It can also be thought of as a steep grade when driving parallel to the y-axis, but flat when driving parallel to the x-axis.

#### Cankur

Hello

I am trying to solve a problem that involves "the plane y=z". How exactly should I think of this plane? It's obviously one of three planes that can be formed between the three diffirent axes, but which one? What is the intuition behind "y=z"?

A really basic question to some of you, I bet. Any help would be appreciated. The understanding behind it all is what I really want to achieve (not just the right answer).

Thanks!

In the y-z plane draw a line of 45 degrees. This line should be the cross section of the plane you want to imagine.

As Anshuman said, first draw a yz-coordinate plane and draw the line y= z (through the origin, at 45 degrees to the axes. Now imagine the x-axis coming out of the paper toward you and the plane being that line coming straight out.
For example, the plane contains the points (x, a, a) for any x or a.

another way to think of this plane, is that it consists of all vectors of the form:

x(1,0,0) + y(0,1,1), for any real numbers x,y.

that is, P = span({(1,0,0), (0,1,1)}).

this plane intersects the "x-axis" (the line x(1,0,0)) and the line y = z. (if we are looking perpendicular to the x-axis (from the "negative part" where x < 0, so the positive numbers are ahead of us) at the yz-plane, we would see our plane "tilted" 45 degrees to the left. if we were on the "positive side" of the x-axis, perpendicluar to it and looking towards the origin, we would see our plane as titled 45 degrees to the right).

driving up our plane parallel to the y-axis it's a rather steep grade, but if we make a left or right turn, and go parallel to the x-axis, it's perfectly flat (unless it's icy, in which case we'll slip down it sideways).