# Vector Function of Cone & Plane Intersection Curve

• Litcyb
In summary, the problem asks for a vector function that represents the curve of intersection between the cone z = sqrt( x^2 + y^2) and the plane z = 1+y. Using x as the parameter, the solution is x=t; y=(t^2-1)/2; z=(t^2+1)/2. If y is used as the parameter, the solution is y=t; x=(2t+1)^(1/2); z=t+1. However, the second solution only represents part of the curve and excludes points with negative values of x. To include all points, the solution x=(-(2t+1)^(1/2)) would also need to be used
Litcyb

## Homework Statement

Find a vector function that represents the curve of intersection of the two surfaces:
The cone z = sqrt( x^2 + y^2) and the plane z = 1+y.

## Homework Equations

z = sqrt( x^2 + y^2) and the plane z = 1+y.

## The Attempt at a Solution

This problem can be solved as following using x as the parameter.
x^2+y^2 = z^2 = (1+y)^2 = 1+2y+y^2. => x^2 = 1 + 2y.

x=t; y = (t^2-1)/2; z = 1+(t^2-1)/2 = (t^2+1)/2

My question is, what if we use y as the parameter,

i get ,

y=t, x=(2t+1)^(1/2) z=t+1,

Last edited by a moderator:

Litcyb said:

## Homework Statement

Find a vector function that represents the curve of intersection of the two surfaces:
The cone z = sqrt( x^2 + y^2) and the plane z = 1+y.

## Homework Equations

z = sqrt( x^2 + y^2) and the plane z = 1+y.

## The Attempt at a Solution

This problem can be solved as following using x as the parameter.
x^2+y^2 = z^2 = (1+y)^2 = 1+2y+y^2. => x^2 = 1 + 2y.

x=t; y = (t^2-1)/2; z = 1+(t^2-1)/2 = (t^2+1)/2

My question is, what if we use y as the parameter,

i get ,

y=t, x=(2t+1)^(1/2) z=t+1,

Sure, with the proviso that it only represents part of the curve. For example, points with negative values of x won't appear in the second parametrization (where you also should specify t>=(-1/2)). You'd need the x=(-(2t+1)^(1/2)) solution as well to get them all.

## 1. What is a vector function of cone and plane intersection curve?

The vector function of cone and plane intersection curve is a mathematical representation of the points where a cone and a plane intersect in three-dimensional space. It describes the position of each point on the intersection curve using a vector, which has both magnitude and direction.

## 2. How is the vector function of cone and plane intersection curve calculated?

The vector function of cone and plane intersection curve is calculated by finding the points where the equations of the cone and plane are equal. These points can then be represented as a vector in terms of a parameter, such as t or θ, which varies along the curve.

## 3. What information can be obtained from the vector function of cone and plane intersection curve?

The vector function of cone and plane intersection curve provides information about the position, direction, and curvature of the intersection curve. It can also be used to calculate the length, area, and volume of the curve, as well as its relationship to the original cone and plane.

## 4. How is the vector function of cone and plane intersection curve used in real-life applications?

The vector function of cone and plane intersection curve has various applications in fields such as engineering, physics, and computer graphics. It is used to model and analyze the behavior of objects in three-dimensional space, such as the trajectory of a projectile or the shape of a 3D surface.

## 5. What are some challenges involved in working with the vector function of cone and plane intersection curve?

Some challenges in working with the vector function of cone and plane intersection curve include finding the correct equations for the cone and plane, solving for the points of intersection, and determining the appropriate parameter to use. Additionally, the curve may have complex or discontinuous parts, which can make it difficult to visualize and analyze.

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