Problem 1 - Calculus Early Transcendentals

Question

Give two pieces of information which, taken together, uniquely determine a plane.

Step-by-Step Solution

Answer

Answer: The unique plane determined by the given point P(2, 3, 4) and normal vector N(1, -1, 2) is defined by the equation: x - y + 2z = 7.

Step 1: Choose a Point on the Plane

Select a point (x1, y1, z1) that lies on the plane. This point will be used together with the normal vector to define the plane within the Cartesian coordinate system. For example, let's consider the point P(2, 3, 4) on the plane.

Step 2: Choose a Normal Vector to the Plane

Select a vector N = (a, b, c) that is normal to the plane. This vector must be perpendicular to the plane's surface. For instance, let's consider a given normal vector N = (1, -1, 2).

Step 3: Write the Equation of the Plane

Now that we have a point P(x1, y1, z1) on the plane and a normal vector N(a, b, c), we can write the equation of the plane using the following formula: \((a(x - x1) + b(y - y1) + c(z - z1)) = 0\) By plugging our chosen point P(2, 3, 4) and normal vector N(1, -1, 2) into the formula, we get the equation of the plane: \((1(x - 2) - 1(y - 3) + 2(z - 4)) = 0\)

Step 4: Simplify the Equation of the Plane

Simplify the equation by expanding and combining like terms: \((1(x - 2) - 1(y - 3) + 2(z - 4)) = 0\) \((x - 2 - y + 3 + 2z - 8) = 0\) \((x - y + 2z - 7) = 0\) The unique plane determined by the given information (point P and normal vector N) is defined by the equation: \(x - y + 2z = 7\)

Key Concepts

Normal Vector

A normal vector is crucial in defining a plane in three-dimensional space. It is denoted by the vector \( \mathbf{N} = (a, b, c) \). This vector is perpendicular to the surface of the plane, which means that it points directly away from the plane at a 90-degree angle. Because of its perpendicular relationship, knowing the normal vector allows you to uniquely specify the plane's orientation in space.

In the context of defining a plane, once a point on the plane is chosen, the normal vector serves as a foundational piece of information. It ensures that given any two distinct parallel planes, it is the normal vector that differentiates their exact standing in space.

Making use of a normal vector means that complex surfaces can be simplified, allowing one to create linear equations that easily describe the full expanse of a plane. This vector basically captures the tilt or slant of the plane. Thus, in plane geometry, the normal vector is not just about direction but also plays a pivotal role in mathematical equations that relate to the plane and its properties.

Cartesian Coordinate System

The Cartesian coordinate system, characterized by its three basic axes—x, y, and z—is a structural backbone in geometry, providing a clear framework for describing the location of points in a three-dimensional space. It allows us to depict geometric shapes, such as planes, in a structured manner.

In this system, each point in space is described by a set of three numbers, known as coordinates, \( (x, y, z) \). These coordinates enable precise definitions of surfaces like planes, because when paired with a normal vector, they help formulate a linear equation representing a plane. For instance, a point \( P(x_1, y_1, z_1) \) aids in determining a specific plane in conjunction with a normal vector by transforming the abstract concept of a plane into a concrete equation: \( a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \).

This system's beauty lies in its simplicity and symmetry, making geometric interpretations more accessible and linking various mathematical concepts together seamlessly. The Cartesian coordinate system forms the foundation upon which advanced geometrical studies in calculus and other disciplines are built, offering a unified language for articulating and solving spatial problems.

Geometry in Calculus

Geometry in calculus extends traditional geometric principles into the realm of advanced mathematics, enabling the calculation of complex shapes, surfaces, and volumes. Calculus helps refine our understanding of geometry by incorporating the ideas of change and motion.

In relation to planes, calculus uses geometric concepts like that of a plane's equation or the normal vector to solve integrative problems involving surface areas, volumes, and rates of change. For example, when calculating the flux across a surface or optimizing functions subject to spatial constraints, understanding the geometry of the involved planes becomes essential.

In essence, calculus transforms static geometric figures into dynamic systems that can be analyzed and manipulated. This deep integration allows one to not just define a plane but also explore its tendencies and interactions with other surfaces in a multi-dimensional environment. Geometry in calculus thus provides a rich toolset for tackling real-world problems that delve into the structure and shape of different physical spaces or objects.