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.
$$
x^{2}+y^{2}+z^{2}=4, \quad x+z=2 \quad x=1+\sin t
$$

Key Concepts

Vector-Valued Function

A vector-valued function describes a curve by assigning a vector (with components corresponding to spatial coordinates) to every value of a parameter. This form is particularly effective for handling curves in space as it consolidates the x, y, and z components into one coherent mathematical object, facilitating further calculations like differentiation and integration.

Parametric Representation

Parametrization involves expressing the coordinates of points on a curve as functions of a parameter, simplifying the analysis of complex curves. By setting one coordinate, such as x = 1 + sin t, one can express the remaining coordinates in terms of t, turning an implicit relation into an explicit set of functions.

Intersection of Surfaces

This concept involves finding the common set of points that satisfy the equations of two or more surfaces. In multivariable calculus, obtaining the intersection curve of surfaces like a sphere and a plane is key to understanding spatial relationships and forms the basis for further analysis or parameterization.

Sphere Equation

A sphere's equation, such as x² + y² + z² = 4, defines a three-dimensional surface where every point is at a constant distance (here, 2 units) from a center point (the origin). Recognizing this helps in visualizing the object and understanding how it might intersect with other geometric surfaces.

Plane Equation

A plane equation like x + z = 2 represents an infinite flat surface in three-dimensional space. This concept is critical, as it demonstrates how a plane can intersect with other surfaces (for example a sphere) to produce curves, often circles or ellipses, depending on the relative positions and angles.

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