3 dimension coordinate systems

[roadrunner]

Homework Statement

Describe in words the region of R3 (3 dimension) represented by the following inequality.

1)xyz=0

Homework Equations

none i know of

The Attempt at a Solution

no idea where to start. I know that this means one variable must be equal to 0, but i don't know how to classify it.

for example

x^2+y^2=r^2 is for a circle...

x^2+y^2+z^2=r^2 is a sphere

x^2 is a parabola

y=mx+b is a line
how would i clasify xyz... :)


NUMBER 2!

Write an inequality to describe the region.

a) region consisting of the firts octant bounded by z=1 y=2 z=3


what I've tried

i'm nt sure if this means x is bounded by x=0 and x=1...or if it means x<=1 and can continue infinatly (same with y and z)

and also, i think id have to make is such 6x=6 3y=6 2z=6 and make it in the form of x^2+y^2+z^2 >= 6 (or <=6 depending on the bounds)

thanks for some input ! :)

 

[D H]

Can you describe the solution to x=0? To xy=0? Start simple, work your way up.

 

[roadrunner]

x=0 means there is only a y coordinate...so (0,y) and xy=0 means (0,y,z) or (x,0,z) or (0,0,z) so xyz=0 means that (x,y,0) or (x,0,z) or(0,y,z) or (x,0,0) or (0,y,0) or (0,0,z) or (0,0,0)

but how do i relate this to some sort of shape? or would i just use what i did?

 

[D H]

Describe y=constant (or x=constant) in simple terms. Zero is just a special constant.

 

[roadrunner]

so z x and y and all constants...and at least one must be 0? that's how i would describe it?

and how about the 2nd question i had?


NUMBER 2!

Write an inequality to describe the region.

a) region consisting of the firts octant bounded by z=1 y=2 z=3


what I've tried

i'm nt sure if this means x is bounded by x=0 and x=1...or if it means x<=1 and can continue infinatly (same with y and z)

and also, i think id have to make is such 6x=6 3y=6 2z=6 and make it in the form of x^2+y^2+z^2 >= 6 (or <=6 depending on the bounds)

thanks for some input ! :)

 

[D H]

Slow down, roadrunner. You are going too fast and in the wrong direction. You appeared to be having troubles with some very basic concepts, so I started simple. Let's start even simpler, with the equation x=c: What simple geometric shape does x=c represent in R¹, R², and R³? I'll give you a start: In R¹ x=c represents a point.

Now specialize to the special constant zero. In R³, the three geometric shapes defined by three equations x=0, y=0, and z=0 have special names.

Finally, what does xyz=0 mean?

Problem 2. What simple expression defines the first octant (use the standard definition here)? This is a simple relation on (x,y,z). Now all you have to do is shift this so that the reference point is not the origin.

 

[roadrunner]

in R2 x=c means just a point also, and in R3?

quesiton two

X>=1 Y>=0 Z>=0 defins the firts octant. so would it be (x-1) +(y-2) + (y-3) >=0? that seems wrong, because that sitll doesn't show that x y and z have individual restrictions.

edit!

just noticed book said inequalities so can i just go x>=1 y>=2 z>=3

 

[D H]

in R2 x=c means just a point also, and in R3?

Not so fast, grasshopper. What simple geometric shape does y=mx+b represent in R²?

 

[roadrunner]

a line
but i thought you said just x=c?
did you mean in R2 use y=mx+b and in R3 use another formula that relates x z y?

 

[D H]

Start with y=mx+b. Its a line in R². Now set m to zero and change that 'b' to c. You get y=0*x+c, or simply y=c. This is still a line. Now change the 'c' to zero. By the way, this line (y=0) has a special name. x=my+b is also an equation of a line. Now set m to zero and b to zero. This (x=0) is still a line, and this line has a special name also.

To recap, in R¹, the equation x=c represents a point, and in R² it represents a line. What does this equation represent in R³?

 

[roadrunner]

parabola?

and the special name being the axis?

 

[D H]

Stop guessing. How could it possibly be a parabola? A parabola "lives" in a plane, not in three dimensions. What is the next step in the progression a point, a line, a ... ?

Someone else needs to help roadrunner here, I am off 'til tomorrow morning.

 

[roadrunner]

circle?
i said parabola cause c=point x=line so i assume enxt was x^2 which is a parabola...but it would be a circle

 

[roadrunner]

ooo maybe at something? would xyz=0 refer to any point on an axis?

 

[HallsofIvy]

A very "relevant equation" is that if ab= 0 then either a= 0 or b= 0. If xyz= 0, then at least one of x, y, or z is 0. No, it is not "any point on an axis" because points on axes have two coordinates 0.

 

[roadrunner]

so then would it be any point on the plane xy, xz, yz ?

 

[D H]

Bingo.

 

[roadrunner]

thanks man! haha