Question

In Exercises \(19-26,\) find a vector-valued function whose graph is the indicated surface. The plane \(z=y\)

Short answer

The vector-valued function representing the plane \(z=y\) is \(r(u,v) = \).

Step-by-step solution

Identify the given plane equation

The surface is given by the plane equation \(z = y\). In this plane equation, z-coordinate is equal to the y-coordinate.

Express the plane equation in vector function form

To represent the plane as a vector function, each component (x, y, z) of the vector can be expressed as a function of two variables (say u and v). For simplicity, consider \(u = x\) and \(v = y\). Now, substitute \(v = y\) in the equation \(z = y\) to get \(z = v\). Thus, the vector function is given by \(r(u,v) = \).

Write the final vector-valued function for the plane

Putting it all together, the vector function representing the given plane is \(r(u,v) = \).

Key concepts

Plane Equation

Understanding the plane equation is fundamental when dealing with three-dimensional geometry. The general form of a plane equation is given by \( Ax + By + Cz + D = 0 \), where A, B, C, and D are constants that determine the orientation and position of the plane in three-dimensional space.

When you're provided with a specific form like \( z = y \), it's a simplified plane equation, indicating that the plane's elevation (z-coordinate) is directly equal to its north-south position (y-coordinate). This particular plane is vertical with respect to the x-axis and makes a 45-degree angle with both the y and z-axes. This kind of straightforward equation is very practical for constructing a vector-valued function which represents the plane in a more algebraically manipulatable form, especially useful in calculus and physics applications.

Parametric Equations

Parametric equations express geometric and algebraic relationships using parameters. Unlike traditional equations that provide a direct relationship between variables, parametric equations introduce one or more additional variables—parameters—to define the relationships more flexibly.

In the context of a plane like \( z = y \), parametric equations allow us to express each coordinate (x, y, z) as functions of two independent parameters, typically referred to as u and v. The advantage is that parametric equations describe an infinite set of points on the surface by varying the parameters, which helps in graphing the surface or calculating integrals over the plane in calculus.

Vectors in Calculus

Vectors are directional quantities with both magnitude and direction, and they play a crucial role in calculus, especially when extending the study into three dimensions. Calculus with vectors, often referred to as vector calculus, allows us to analyze space curves and surfaces.

A vector-valued function, such as \( r(u,v) = \), compactly represents a surface by assigning a vector to each point determined by the parameters u and v. These vectors can be operated on using calculus tools such as differentiation and integration to study the properties of the surface, including tangent planes, normal lines, and curvature. Vector calculus is essential in fields like physics, where it's used to model phenomena like electromagnetic fields and fluid dynamics.