Sketch the space curve represented by the intersection of...

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter.
Surfaces $\quad$ Parameter
$x^{2}+y^{2}+z^{2}=10, \quad x+y=4 \quad x=2+\sin t$

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Key Concepts

Parameterization

Parameterization is the process of representing a curve or surface through one or more parameters. In the context of space curves, parameterization converts the geometric description into an algebraic form, making it easier to perform calculations such as differentiation, integration, and sketching.

Vector-Valued Functions

A vector-valued function expresses a coordinate vector in terms of a single parameter, typically written as r(t) = <x(t), y(t), z(t)>. This approach simplifies the analysis and visualization of curves in space by reducing the problem to a one-dimensional parameter domain.

Space Curve

A space curve is a one-dimensional locus of points in three-dimensional space. These curves can arise from the intersection of surfaces or as the image of a vector-valued function, and they provide a geometric representation of paths or trajectories in space.

Intersection of Surfaces

This concept involves finding the set of points that satisfy the equations of two or more surfaces simultaneously. The intersection typically forms a curve in three-dimensional space, and identifying this curve requires solving a system of equations to reduce the degrees of freedom involved.

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