Problem 26 - Thomas Calculus

Question

Find equations for the planes in Exercises 21–26. The plane through \(A(1,-2,1)\) perpendicular to the vector from the origin to \(A\)

Step-by-Step Solution

Answer

The plane equation is \( x - 2y + z = 6 \).

Step 1: Identify the given point

The plane passes through the point \( A(1, -2, 1) \). This is the point given on the plane.

Step 2: Determine the normal vector

Since the plane is perpendicular to the vector from the origin to \( A \), use this vector as the normal. The vector is \( \mathbf{n} = \langle 1, -2, 1 \rangle \).

Step 3: Recall the equation of a plane

The general equation of a plane is given by \( ax + by + cz = d \), where \( \langle a, b, c \rangle \) is the normal vector and \( (x_0, y_0, z_0) \) is a point on the plane.

Step 4: Substitute into the plane equation

Substitute \( a = 1 \), \( b = -2 \), and \( c = 1 \), and the point \( A(1, -2, 1) \) into the equation of the plane: \[ 1(x - 1) - 2(y + 2) + 1(z - 1) = 0 \]

Step 5: Simplify the equation

Expand and simplify the equation obtained in Step 4: \[ x - 1 - 2y - 4 + z - 1 = 0 \]This simplifies to \[ x - 2y + z = 6 \].

Key Concepts

Vector Analysis

Vector analysis is an essential part of mathematics and physics, playing a crucial role in understanding the direction and magnitude of quantities within a 3D space. In simple terms, vectors are objects that have both size and direction, represented as arrows in a coordinate system. They help us describe the orientation and movement within three-dimensional spaces. Vectors are often denoted as coordinates, such as \( \langle x, y, z \rangle \), indicating their endpoint based on the origin of a graph.
Applications of vector analysis are vast, enabling us to model phenomena like force, velocity, and displacement. In the exercise, the vector from the origin to point \( A(1, -2, 1) \) is critical because it serves as the basis for defining other characteristics of the plane, such as the normal vector. Understanding how to manipulate and calculate vector properties allows us to explore and describe the physical and geometric context accurately.

Normal Vector

The normal vector is a concept central to the geometry of planes. It is a vector that is perpendicular to a surface at a given point. This perpendicular nature makes it extremely useful for defining planes in 3D spaces. In equations, the components of the normal vector \( \langle a, b, c \rangle \) appear as coefficients in the plane's equation \( ax + by + cz = d \).
In our exercise, the normal vector \( \mathbf{n} = \langle 1, -2, 1 \rangle \) is derived from the vector pointing towards point \( A(1, -2, 1) \) from the origin. This means the plane we are working with is oriented around this vector. The perpendicular direction of the normal vector is what allows us to define the plane's orientation uniquely in 3D space. It ensures that any kind of rotation or tilt of the plane can be easily articulated.

Plane Geometry

Plane geometry involves studying flat surfaces, which can be either one-dimensional or exist in 3D space. When we speak about planes in three-dimensional geometry, it is a flat, two-dimensional surface extending infinitely. The mathematical description of a plane is key to representing such surfaces, and one of the common forms of this equation is \( ax + by + cz = d \).
This equation relies on the normal vector and a specific point through which the plane passes. By substituting the coordinates of a point such as \( A(1, -2, 1) \) into the equation, along with the components of the normal vector, we can define the plane's location and orientation within the 3D space. Plane geometry is foundational for working on more complex geometrical structures and for providing a framework for visualizing problems within mathematics and physics.

3D Coordinate System

The 3D coordinate system is a method used to pinpoint the location of points in space through an ordered triplet of numbers. Each point is specified using three coordinates: \( (x, y, z) \), representing the respective distances along the axes of the space. This system allows us to navigate and represent three-dimensional spaces in mathematical terms efficiently.
Understanding how to use this system is crucial for solving problems that involve spatial designs and positions, as seen in the exercise. For instance, the point \( A(1, -2, 1) \) needs to be precisely located within the 3D space to apply it correctly in the plane equation. The 3D coordinate system is extensively utilized in fields ranging from computer graphics to physics, offering a standardized way to communicate spatial data.