Problem 65 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P .\) $$ \mathbf{n}=2 \mathbf{i}-4 \mathbf{j}+3 \mathbf{k} ; P(1,2,-3) $$

Step-by-Step Solution

Answer

The equation of the plane is \( 2x - 4y + 3z = -15 \).

Step 1: Identify the Given Information

We are given a normal vector \( \mathbf{n} = 2\mathbf{i} - 4\mathbf{j} + 3\mathbf{k} \) and a point \( P(1, 2, -3) \) through which the plane passes. The normal vector's components are \( A = 2 \), \( B = -4 \), and \( C = 3 \).

Step 2: Set Up the Plane Equation

The general equation of a plane is \( Ax + By + Cz = D \). With \( A = 2 \), \( B = -4 \), and \( C = 3 \), we substitute these values into the equation, yielding \( 2x - 4y + 3z = D \).

Step 3: Use the Point to Find D

Substitute the coordinates of point \( P(1, 2, -3) \) into the plane equation: \[ 2(1) - 4(2) + 3(-3) = D \] Calculate step-by-step: \( 2 - 8 - 9 = D \), which simplifies to \( D = -15 \).

Step 4: Write the Final Equation of the Plane

Replace \( D \) in the plane equation with \( -15\) to get the equation of the plane: \[ 2x - 4y + 3z = -15 \].

Key Concepts

Normal Vector

In the context of 3D geometry and plane equations, a normal vector is a vector that is perpendicular to a plane. Think of it as the arrow that sticks out directly from any flat surface. Mathematically, it is represented by its components along the x, y, and z axes. For example, the normal vector \( \mathbf{n} \) in our exercise is \( 2\mathbf{i} - 4\mathbf{j} + 3\mathbf{k} \), which means:

• It moves 2 units in the direction of the x-axis.
• It moves -4 units in the direction of the y-axis.
• It moves 3 units in the direction of the z-axis.

The components \( A = 2 \), \( B = -4 \), and \( C = 3 \) form the critical connection between the normal vector and the plane's equation. These coefficients determine the plane's orientation in 3D space. Using these, we form an equation that includes any point on the plane, ensuring it meets the perpendicular distance criteria defined by the normal vector.

3D Geometry

3D Geometry, also known as three-dimensional geometry, involves the study of shapes and figures that have three dimensions: length, breadth, and height. Unlike the simpler 2D world, where we only deal with x and y axes, 3D geometry includes an added dimension - the z-axis.
Understanding the layout in 3D requires visualizing how points, lines, and planes nest within this space. Each point in 3D space can be described with three coordinates \( (x, y, z) \).
Planes in 3D space can be identified by their equation \( Ax + By + Cz = D \). Here, \( (A, B, C) \) are part of the normal vector that determines the plane's slant, while \( D \) is derived based on specific points on the plane.In practical terms, these concepts allow us to understand how objects exist and relate to one another in space, forming the basis for much of physical modeling, CAD designs, and real-world simulations.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is the study of geometric shapes using a coordinate system. In 3D, this system is extended to accommodate three axes: x, y, and z.
When working with planes in coordinate geometry, the position of a plane is defined using both a normal vector and a specific point. For our example exercise, the plane passes through the point \( P(1, 2, -3) \). This tells us exactly where the plane is located in the vast 3D space.
The equation of a plane itself, \( Ax + By + Cz = D \), is a direct application of coordinate geometry principles. In this formula:

• \( A, B, C \) are coefficients obtained from the normal vector, determining direction.
• \( D \) is the constant calculated using a known point on the plane.

Through plotting such equations, one can visualize spatial relationships, tackle engineering problems, and explore aircraft and architecture designs. Coordinate geometry thus serves as a bridge between algebra and geometry, making complex ideas about shapes and spaces more accessible.