How Do I Understand Octants and Graphs in Calculus III?

[omohiuddin]

I have a few questions on a problem in my first Calc III class. "The solid cube in the first octant bounded by the coordinate planes and x=2,y=2 and z=2, write the inequality to describe the set". I am new to the 3 dimensional cartesian system, but i still do not get the concept of the octants because i figured their would be 12, 4 for each plane. and also,
x^2 + Y^2 <=1 and no restriction on Z, how does this graph look like. I have been having trouble with those 2 problems. I would greatly apreciate your help.

 

[symbolipoint]

In two dimensions on the cartesian plane, you have four quadrants. When you include the third dimension, you may call the number line the z axis. This z axis points above and below the x y plane; above the x y plane will be four octants and below the x y plane will be another four octants. 4 + 4 = 8.

 

[nicktacik]

Let's look at all the possibilities for (x,y,z)

(+++)
(++-)
(+-+)
(+--)
(-++)
(-+-)
(--+)
(---)

That makes 8, hence octant.If you have a vector valued function (cost, sint, t), it's called a helix.
This is what it looks like:
http://www.math.rutgers.edu/~greenfie/mill_courses/math251/gifstuff/helix_tan_line.gif

 

[HallsofIvy]

I have a few questions on a problem in my first Calc III class. "The solid cube in the first octant bounded by the coordinate planes and x=2,y=2 and z=2, write the inequality to describe the set". I am new to the 3 dimensional cartesian system, but i still do not get the concept of the octants because i figured their would be 12, 4 for each plane.

But you are not restricted to the coordinate planes. In the xy-plane you have 2 coordinate axes each having 2 sides: they divide the plane into 2²= 4 quadrants. In 3 dimensions, you have 3 coordinate axes that divide the entire space into 2³= 8 "octants". You can also, as nicktacik said, look at the signs: each of x, y and z can be either positive or negative: again 2³= 8 possible combinations

and also,
x^2 + Y^2 <=1 and no restriction on Z, how does this graph look like. I have been having trouble with those 2 problems. I would greatly apreciate your help.

In the plane, x^2+ y^2\le 1 is the unit disk. Since z can be anything imagine the disk moving straight up and down: you have the inside of an infinitely long cylinder.