Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the plane passing through \((-4,-1,2)\) and parallel to (a) the \(x y\) -plane (b) the plane \(2 x-3 y-4 z=0\)

Step-by-Step Solution

Answer

(a) Plane equation: \(z = 2\), (b) Plane equation: \(2x - 3y - 4z + 13 = 0\).

Step 1: Identify the Normal Vector for the x-y Plane

The normal vector for the plane parallel to the xy-plane is along the z-axis. Hence, the normal vector is \( \mathbf{n} = (0, 0, 1) \).

Step 2: Use Point-Normal Form for the Plane Parallel to xy-Plane

For a plane with a known normal vector \((a, b, c)\) passing through a point \((x_0, y_0, z_0)\), the equation is: \( a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \). Substituting \((0, 0, 1)\) for \(\mathbf{n}\) and \((-4, -1, 2)\) for \((x_0, y_0, z_0)\), we derive the equation \( 0(x + 4) + 0(y + 1) + 1(z - 2) = 0 \). Simplifying gives \( z = 2 \).

Step 3: Identify the Normal Vector for the Given Plane

The given plane is \(2x - 3y - 4z = 0\) with normal vector \( \mathbf{n} = (2, -3, -4) \). Planes parallel to this will have the same normal vector.

Step 4: Use Point-Normal Form for the Plane Parallel to Given Plane

Using the same point-normal form as before, substitute \((2, -3, -4)\) for \(\mathbf{n}\) and the point \((-4, -1, 2)\): \( 2(x + 4) - 3(y + 1) - 4(z - 2) = 0 \).

Step 5: Simplify the Plane Equation

Distribute and simplify: \( 2x + 8 - 3y - 3 - 4z + 8 = 0 \). This simplifies further to \( 2x - 3y - 4z + 13 = 0 \).

Key Concepts

Point-Normal Form

The point-normal form of a plane is a powerful method for finding a plane's equation, especially when we know a point on the plane and its normal vector. A plane's equation in the point-normal form can be written as:

• For a plane with a normal vector \(a, b, c\) passing through a point \(x_0, y_0, z_0\), the equation is: \(a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\).

This form is essential as it illustrates how the orientation of the plane relates to the three-dimensional coordinate system.
This form demonstrates how any point \(x, y, z\) on the plane will satisfy the equation when substituted into it.
The value of the products inside the parentheses equals zero, showing that these points lie on a plane as they mutually align perpendicularly to the normal vector.

Normal Vector

A normal vector is a vector that is perpendicular to a surface or a plane. It is crucial for defining the orientation of a plane as it points in the direction away from the plane.

• For the xy-plane, the normal vector is \(0, 0, 1\), meaning it points directly upwards along the z-axis.
• Any plane parallel to another will share the same normal vector.

A plane parallel to another plane, such as the xy-plane, has a normal vector that does not change its direction, only its position.
The direction of the normal vector completely governs how the parallel plane is situated in space, while still sharing its foundational alignment.

XY-Plane

The xy-plane is a fundamental concept in geometry. This plane is essentially a horizontal flat surface extending infinitely in all directions along the x and y axes, with the z coordinate fixed at zero.

• The equation \(z = 0\) represents the xy-plane.
• Any point on this plane will have a z-value of zero, specifying its lack of height from the baseline.

This plane serves as a reference for many 3D calculations, including determining parallelism in space.
Any plane that is parallel to the xy-plane will have a constant z value, which makes calculating its equation straightforward by focusing on the fixed z-coordinate.

Plane Parallelism

Plane parallelism is when two planes never intersect each other, no matter how far they are extended in space. This occurs when their normal vectors are identical, indicating they are parallel.

• If two planes have the same normal vector, they will always be parallel.
• Parallel planes do not share points other than at a potential infinite distance.

This constant alignment is crucial for specific applications, such as architectural designs or certain computational models.
Recognizing two planes as parallel allows for the simplification of many mathematical models by reducing the nature of interactions between objects or surfaces into consistent guidelines based on their constructed normals.