Find parametric equations for the line tangent to the curve...

Find parametric equations for the line tangent to the curve of intersection of the surfaces at the given point.
$$\begin{aligned}
&\text { Surfaces: } x^{2}+2 y+2 z=4, \quad y=1\\
&\text { Point: } \quad(1,1,1 / 2)
\end{aligned}$$

Key Concepts

Gradient Vector

The gradient vector of a function is composed of its partial derivatives and points in the direction of the greatest rate of increase of the function. For surfaces defined implicitly by an equation, the gradient is perpendicular to the surface, making it a key tool in analyzing directions and finding normals.

Curve of Intersection

A curve of intersection is formed by the set of points that satisfy the equations of two or more surfaces simultaneously. Understanding this concept is crucial because the resulting curve embodies the shared geometry of the surfaces, and many problems in multivariable calculus involve finding its properties, such as tangents and normals.

Tangent Line

The tangent line at a point on a curve is the line that best approximates the curve near that point. It reflects the instantaneous direction of the curve and is fundamental in analyzing the curve's local behavior, particularly when finding rates of change along the curve.

Parametric Equations

Parametric equations express a curve or line by defining each coordinate as a function of one or more parameters. This representation is especially useful for describing curves in space, as it allows for a straightforward description of the position of points along the curve in terms of a continuously varying parameter.

Cross Product

The cross product is a vector operation that takes two vectors in three-dimensional space and returns a third vector which is perpendicular to both. In the context of curves defined by the intersection of two surfaces, taking the cross product of the gradients of the surfaces yields a vector that is tangent to the intersection curve, as it is orthogonal to both surface normals.

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