Intersection of a sphere and plane

[yy205001]

Homework Statement

Show that the circle that is the intersection of the plane x + y + z = 0 and the
sphere x² + y² + z² = 1 can be expressed as:

x(t) = [cos(t)-sqrt(3)sin(t)]/sqrt(6)
y(t) = [cos(t)+sqrt(3)sin(t)]/sqrt(6)
z(t) = -[2cos(t)]/sqrt(6)

Homework Equations

The Attempt at a Solution

At first, I tried to let z=-x-y then sub it into the equation of the sphere:
x²+y²+(-x-y)²=1
x²+y²+x²+2xy+y²=1
2x²+2xy+2y² = 1
And i stuck at here, I can't make it into a circle equation.

So, I changed another way.
I equated two equations, so it becomes:
x² + y² + z² -1 = x+y+z
x²+ x + y² + y + z² +Z -1 =0
(x-1/2)² + (y-1/2)² + (z-1/2)² = 7/4 *By completing the square

But isn't this is an equation of a sphere??
Shouldn't it just a circle?

Any help is appreciated!

 

[tiny-tim]

hi yy205001!

Show that the circle that is the intersection of the plane x + y + z = 0 and the
sphere x² + y² + z² = 1 can be expressed as:

x(t) = [cos(t)-sqrt(3)sin(t)]/sqrt(6)
y(t) = [cos(t)+sqrt(3)sin(t)]/sqrt(6)
z(t) = -[2cos(t)]/sqrt(6)

why don't you just bung the three parameter formulas into the two original equations, and show that they work?

I equated two equations, so it becomes:
x² + y² + z² -1 = x+y+z

but that also gives you the intersections of x + y + z = k and the
sphere x² + y² + z² = 1 + k, for all values of k !

 

[LCKurtz]

hi yy205001!


why don't you just bung the three parameter formulas into the two original equations, and show that they work?

That would show the parametric curve lies on the circle. But is it the whole circle? Looks like there is more to do.

 

[yy205001]

And I am thinking is there anything do with the normal vector of the plane.

 

[tiny-tim]

what does that have to do with the parametric equations?

 

[yy205001]

what does that have to do with the parametric equations?

So, I just sub in those 3 parametrize equations into the equation for the plane and sphere, then show they satisfy x²+y²+z²=1 and x+y+z=0?

 

[tiny-tim]

yes

the question starts "show", so all you need to do is show that the answer is correct, you don't have to find the answer from scratch

(and, as LCKurtz says, you'll also have to prove that that parametrisation gives the whole intersection)

 

[yy205001]

I got it!
tiny-tim, LCKurtz, thank you so much for the help!