Question

Find a vector function that represents the curve of intersection of the paraboloid and the cylinder. Use the variable t for the parameter.

Answer 1

The vector function that represents the curve of intersection of the paraboloid and the cylinder, using the parameter (t), is given by
`\( \mathbf{r}(t) = \langle t, t^2, √(1 - t^2) \rangle \)`.

Step-by-step explanation:

To find the vector function representing the curve of intersection between the paraboloid and the cylinder, we consider the equations of both surfaces. Let the paraboloid be represented by
`\( z = x^2 + y^2 \)` and the cylinder by
`\( x^2 + y^2 = 1 \)`. Setting these two equations equal to each other, we get
`\( x^2 + y^2 = z \)` and
`\( x^2 + y^2 = 1 \)`. These equations describe the intersection curve.

Now, parameterize the curve using a parameter (t) to obtain
`\( x(t) = t, \) \( y(t) = t^2, \) and \( z(t) = √(1 - t^2) \)`. Combining these into a vector function, we have
`\( \mathbf{r}(t) = \langle t, t^2, √(1 - t^2) \rangle \)`. This vector function traces the curve of intersection for different values of (t).

The parameter (t) allows us to trace the curve as (t) varies. The resulting vector function
`\( \mathbf{r}(t) \)` provides a convenient way to represent the curve in three-dimensional space. It's important to note that this is a specific solution for the given surfaces, and the parameterization may vary for different pairs of surfaces.

Answer 2

The student is seeking a vector function that describes the curve of intersection between a paraboloid and a cylinder. Assuming general forms for these surfaces, the vector function for the curve of intersection, when the cylinder is centered about the z-axis, is
`\( \vec{r}(t) = \langle r*cos(t), r*sin(t), r² \rangle`, where 't' is the parameter and 'r' the radius of the cylinder.

Step-by-step explanation:

Finding a Vector Function for the Curve of Intersection

The student is asking for a vector function that represents the curve of intersection of two surfaces: a paraboloid and a cylinder. Since the equation for each surface is not specified in the question, we'll assume general forms for both. A paraboloid can be represented by z = x² + y², and a cylinder centered around the z-axis can be represented by x² + y² = r², where 'r' is the radius of the cylinder. To find the curve of intersection, we set the equations equal to each other due to the similarity in their x and y terms, which leads to z = r².

To represent this curve with a vector function, we can parametrize it using trigonometric functions for x and y to satisfy the cylinder's equation. By letting x(t) = r*cos(t) and y(t) = r*sin(t), where 't' is the parameter, we consequently have z(t) = r². Therefore, the vector function that represents the curve of intersection is
`\( \vec{r}(t) = \langle r*cos(t), r*sin(t), r² \rangle`.

To express the answer in a coordinate system with the origin as the center of the cylinder, no further action is necessary because the vector function is already centered around the origin.