Why Can't the Positive Cone Be a Submanifold in R^3?

[Pietjuh]

For a homework assignment i was asked to proof that the positive cone {x^2 + y^2 = z^2, z>= 0} cannot be a submanifold of any dimension of R^3.

It apparently goes wrong at the origin. I guess it's because you can't really speak of a tangent space at that point. So I tried to prove by contradiction you can't have a tangent space at that point. But I couldn't really arrive at a contradiction

Could someone give me a hint?

 

[amcavoy]

For a homework assignment i was asked to proof that the positive cone {x^2 + y^2 = z^2, z>= 0} cannot be a submanifold of any dimension of R^3.
It apparently goes wrong at the origin. I guess it's because you can't really speak of a tangent space at that point. So I tried to prove by contradiction you can't have a tangent space at that point. But I couldn't really arrive at a contradiction
Could someone give me a hint?

Try finding an equation for a tangent plane at the origin. That's the only thing I can think of (which is basically what you already had in mind!)

Alex