# Representing a graph by a Vector-Valued Function

## Homework Statement

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.

## Homework Equations

z = x^2 + y^2, x + y = 0, x = t

## The Attempt at a Solution

Space curve sketched (elliptic paraboloid corresponding to z-axis)
Vector valued function: x = t, y = -t; z = t^2 + (-t)2; z = 2t^2; r(t) = ti - tj + 2t^2k

**Intersection of the surface, not sure how to obtain this. I have the feeling once I get it I'm gonna be shaking my head for having forgotten something. So I've tried setting
x^2 + y^2 = x + y
tried substituting in t for x and y values
tried reverse engineering what I'm supposed to do with the answers plugged in to the equations.
tried finding x int, y int, and z int.

Just don't know how to procede. Any assistance is greatly appreciated.

Dick
Science Advisor
Homework Helper

## Homework Statement

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.

## Homework Equations

z = x^2 + y^2, x + y = 0, x = t

## The Attempt at a Solution

Space curve sketched (elliptic paraboloid corresponding to z-axis)
Vector valued function: x = t, y = -t; z = t^2 + (-t)2; z = 2t^2; r(t) = ti - tj + 2t^2k

**Intersection of the surface, not sure how to obtain this. I have the feeling once I get it I'm gonna be shaking my head for having forgotten something. So I've tried setting
x^2 + y^2 = x + y
tried substituting in t for x and y values
tried reverse engineering what I'm supposed to do with the answers plugged in to the equations.
tried finding x int, y int, and z int.

Just don't know how to procede. Any assistance is greatly appreciated.

If x=t and x+y=0, what is y in terms of t? Now what is z in terms of t? It is really simple.

I already have the vecor valued function. I'm looking for the points of intersection. How do I find that the surfaces intersect at ((root2), -(root2), 4)?

Dick
Science Advisor
Homework Helper
I already have the vecor valued function. I'm looking for the points of intersection. How do I find that the surfaces intersect at ((root2), -(root2), 4)?

You already have the curve that represents the intersection of the two surfaces. The point you give is just one point on that intersection curve. (1,-1,2) is another. There are an infinite number of them, every value of t gives a different one. Are you intersecting that curve with something else that makes t=sqrt(2) special?

ok thanks. Sounds like I wasted a lot of my time, and potentially yours as well as some bandwidth. Thanks again for your time.