How can I visualize the plane y=z?

[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!

 

[Anshuman_]

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.

 

[HallsofIvy]

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.

 

[Deveno]

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).

 

[blue_raver22]

I can provide some insight into visualizing the plane y=z. The equation y=z represents a plane that is parallel to the x-axis and has a slope of 1. This means that for every unit increase in the y-axis, there is also a unit increase in the z-axis. This can be visualized as a flat surface that extends infinitely in the x-direction, with points on the surface having the same value for y and z.

Another way to think about it is to imagine a coordinate system where the x-axis is going into the page, the y-axis is going horizontally, and the z-axis is going vertically. The plane y=z would then be a diagonal line going from the bottom left corner to the top right corner of the coordinate system.

I hope this helps with your understanding of the plane y=z. It's important to remember that visualization is just one way to understand mathematical concepts, but it's equally important to understand the underlying principles and equations that govern them. Keep exploring and asking questions, and you'll continue to deepen your understanding of this and other mathematical concepts.