Calculus III: Vector Functions and Space Curves

**saxyliz** · September 22nd 2008, 05:26 PM

If someone could please help me with this problem it would be most appreciated! I don't even know where to start!

Find a vector function that represents the curve of intersection of the two surfaces (in the 3D space):

The paraboloid: $z = 4x^2 + y^2$
The parabolic cylinder: $y = x^2$

**Jhevon** · September 22nd 2008, 05:29 PM

Originally Posted by saxyliz

If someone could please help me with this problem it would be most appreciated! I don't even know where to start!

Find a vector function that represents the curve of intersection of the two surfaces (in the 3D space):

The paraboloid: $z = 4x^2 + y^2$
The parabolic cylinder: $y = x^2$

plug in $y = x^2$ into $z$ , so that $z = 4x^2 + x^4$

now parametrize the curve using $x = t$ as the parameter

**saxyliz** · September 22nd 2008, 05:37 PM

What exactly do to mean by "parametrize the curve?"

Is it:
$z = 4t^2 + t^4$
?

**Jhevon** · September 22nd 2008, 05:40 PM

Originally Posted by saxyliz

What exactly do to mean by "parametrize the curve?"

Is it:
$z = 4t^2 + t^4$
?

the vector function is $r(t) = \left< x(t), y(t), z(t) \right> = \left< t, t^2 , 4t^2 + t^4 \right>$

**saxyliz** · September 22nd 2008, 05:44 PM

Oh! I get it now! Thanks a lot!

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