Problem 22 - Thomas Calculus

Question

Find equations for the planes in Exercises 21–26. The plane through \((1,-1,3)\) parallel to the plane $$3 x+y+z=7$$

Step-by-Step Solution

Answer

The equation of the plane is \(3x + y + z = 5\).

Step 1: Identify the Normal Vector

The normal vector to the given plane \(3x + y + z = 7\) is \(\mathbf{n} = \langle 3, 1, 1 \rangle\). Since the planes are parallel, the normal vector of our target plane will also be \(\mathbf{n} = \langle 3, 1, 1 \rangle\).

Step 2: Use the Point-Normal Form of a Plane Equation

The equation for a plane given a point \((x_0, y_0, z_0)\) and a normal vector \(\mathbf{n} = \langle a, b, c \rangle\) is \(a(x-x_0) + b(y-y_0) + c(z-z_0) = 0\). Here, the given point is \((1, -1, 3)\) and the normal vector is \(\langle 3, 1, 1 \rangle\).

Step 3: Substitute Point and Normal Vector into Plane Equation

Substituting the point \((1, -1, 3)\) and the normal vector \(\langle 3, 1, 1 \rangle\) into the general form we get: \[3(x - 1) + 1(y + 1) + 1(z - 3) = 0.\] Simplifying, \[3x - 3 + y + 1 + z - 3 = 0\]which simplifies further to:\[3x + y + z = 5.\]

Step 4: Write Down the Final Equation

The equation of the plane parallel to \(3x + y + z = 7\) and passing through \((1, -1, 3)\) is:\[3x + y + z = 5.\]

Key Concepts

Normal Vector

In three-dimensional geometry, a normal vector plays a critical role in defining a plane. A normal vector is a vector that is perpendicular (or orthogonal) to the surface of the plane. When you have the equation of a plane in the form of \(ax + by + cz = d\), the coefficients \(a, b,\) and \(c\) represent the components of the normal vector, \(\mathbf{n} = \langle a, b, c \rangle\).
Understanding the normal vector is key to solving problems involving planes because it provides the direction perpendicular to the plane.

• It helps to determine if two planes are parallel.
• It is used in the point-normal form to find plane equations.

Recognizing that the normal vectors of parallel planes are identical is essential when configuring parallel planes through different points.

Point-Normal Form

The point-normal form of a plane is a method to find the equation of a plane when a point and a normal vector are known. Given a point \((x_0, y_0, z_0)\) and a normal vector \(\mathbf{n} = \langle a, b, c \rangle\), the equation of the plane can be written as:
\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \].
This form helps to establish a direct connection between the geometrical intuition (a point through which the plane passes) and the algebraic description (normal vector defining the direction of the plane).

• It simplifies the creation of plane equations when the normal vector and a point are given.
• Aids in quickly identifying or constructing equations for parallel planes.

Employing the point-normal form enables the calculation of precise plane equations efficiently and accurately.

Parallel Planes

Parallel planes are a fascinating aspect of three-dimensional geometry, characterized by having identical normal vectors. This means that if planes are parallel, their normal vectors will point in the same direction, indicating that the planes never intersect.
It’s important to note the following:

• The distance between parallel planes remains consistent throughout.
• They are absolutely equidistant from each other at any point.
• In the context of plane equations, parallel planes will differ only in the constant terms \(d\), while their other terms remain unvaried.

Understanding parallel planes profoundly affects how we approach problems related to spatial relationships and plane configurations within a three-dimensional space.

Three-Dimensional Geometry

Three-dimensional geometry is a vast field focusing on objects and figures in three-dimensional space, where each point is defined by coordinates \((x, y, z)\). Unlike two-dimensional geometry, three-dimensional geometry explores the relationships between curves, surfaces, and solids in a space where depth adds complexity.
Here are some crucial points about three-dimensional geometry:

• It involves studying planes, vectors, and surfaces in space.
• Concepts such as distance, direction, and angle in space become central.
• Utilizes coordinate systems to define and interpret geometrical figures.

Three-dimensional geometry allows a more comprehensive understanding of spatial relationships, essential for solving advanced geometrical problems and real-world applications. Recognizing the principles of 3D geometry aids in mastering the interactions between different geometrical forms within our three-dimensional world.