nickadams
Dec18-11, 07:51 PM
1. The problem statement, all variables and given/known data
Consider three systems of equations:
x^2 + y^2 + z^2 = 1
y^2 + z^2 = 1
x^2 + y^2 + z^2 = 1
x = 0
y^2 + z^2 = 1
x = 0
Which of these define the same curve and which define different ones?
2. Relevant equations
x^2 + y^2 + z^2 = R^2 is a sphere
x,y, or z = # is a plane
(x,y,z)^2 + (x,y,z)^2 = # is a cylinder
3. The attempt at a solution
I think they all define the same curve; here is why...
Equation 1 of the first system is a sphere centered at the origin with a radius of 1.
Equation 2 of the first system is a cylinder centered around the x axis.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .
Equation 1 of the second system is a sphere centered at the origin with a radius of 1.
Equation 2 of the second system is the yz plane at x=0.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .
Equation 1 of the third system is a cylinder centered at the x axis with a radius of 1.
Equation 2 of the third system is the yz plane at x=0.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .
Am I right?:biggrin:
Consider three systems of equations:
x^2 + y^2 + z^2 = 1
y^2 + z^2 = 1
x^2 + y^2 + z^2 = 1
x = 0
y^2 + z^2 = 1
x = 0
Which of these define the same curve and which define different ones?
2. Relevant equations
x^2 + y^2 + z^2 = R^2 is a sphere
x,y, or z = # is a plane
(x,y,z)^2 + (x,y,z)^2 = # is a cylinder
3. The attempt at a solution
I think they all define the same curve; here is why...
Equation 1 of the first system is a sphere centered at the origin with a radius of 1.
Equation 2 of the first system is a cylinder centered around the x axis.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .
Equation 1 of the second system is a sphere centered at the origin with a radius of 1.
Equation 2 of the second system is the yz plane at x=0.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .
Equation 1 of the third system is a cylinder centered at the x axis with a radius of 1.
Equation 2 of the third system is the yz plane at x=0.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .
Am I right?:biggrin: