Problem 44 - Advanced Engineering Mathematics

Question

In Problems, find an equation of the plane that contains the given point and is perpendicular to the indicated vector. $$ (-1,1,0) ;-\mathbf{i}+\mathbf{j}-\mathbf{k} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

To find the equation of a plane, you need a point on the plane and a normal vector. Here, the point is given as (-1, 1, 0) and the normal vector is (-\mathbf{i}+\mathbf{j}-\mathbf{k}), which is equal to (-1, 1, -1). We will utilize the general equation of a plane: \( ax + by + cz = d \). The coefficients \(a\), \(b\), and \(c\) are derived from the normal vector.

Step 2: Applying the Normal Vector

The normal vector (-1, 1, -1) means that the components \(a\), \(b\), and \(c\) for the plane equation are \(a = -1\), \(b = 1\), and \(c = -1\). This gives the initial equation form of the plane as:\[-x + y - z = d\].

Step 3: Substituting the Point into the Equation

Substitute the point (-1, 1, 0) into the equation:\[ -(-1) + 1 - 0 = d \]. Simplifying gives:\[ 1 + 1 = d \], thus \( d = 2 \).

Step 4: Writing the Final Equation

Now use the values obtained to write the full equation of the plane:\[-x + y - z = 2\].

Key Concepts

Normal Vector

In the context of three-dimensional geometry, a normal vector represents a direction that is perpendicular to a surface. When dealing with planes, this vector is vital. It is orthogonal to every vector that lies on the plane, essentially "pointing" directly out from it.

A normal vector is often made up of component values derived from a geometric problem, which are essential in defining the equation of a plane. For example, the vector given, (-\mathbf{i} + \mathbf{j} - \mathbf{k}), translates to the coordinates given as (-1, 1, -1). These coordinates are key because they correspond to the coefficients (a, b, c) in the general equation of the plane: \[ ax + by + cz = d. \]

When you break down this vector, it provides necessary orientation and slope information. Its role is to ensure that any point-vector calculation involving it will accurately define the relationship of points relative to this space. Thus, the plane will extend in a manner that is exactly perpendicular to this vector.

Point in the Plane

To form an equation of a plane in three-dimensional space, you need a specific point through which the plane passes. This point, denoted as \( (x_0, y_0, z_0) \), acts as an anchor or reference for placing the plane. It is positioned using coordinates that reflect its location within the three-dimensional space.

In our example, the given point is \((-1, 1, 0)\). This means that the plane will pass directly through these coordinates on the 3D grid. By having a fixed point and a normal vector, you can precisely define the plane's location and orientation.

• Each coordinate within the point influences the plane's equation with the substitution \( (x_0, y_0, z_0) \) for \( x, y, z \).
• This step ensures the final equation represents all points lying within the plane, not just the particular one used in calculations.

Each point on the plane satisfies the equation: \[ -x + y - z = 2. \] This equation tells us that wherever the plane stretches, it adheres to this rule, confirming the point's location on this defined surface.

Three-Dimensional Geometry

Three-dimensional geometry expands our exploration beyond flat surfaces into an orientation that has depth in addition to length and width. This space can be a little more challenging to comprehend since it includes a benchmark that directs how we measure and perceive all dimensions.

In this dimensional space, planes are basic building blocks. Planes serve as flat, two-dimensional slices within three-dimensional realms, much like sheets of paper floating in space. A plane is defined with the help of two pivotal components: a point in space and a normal vector.

• The point provides a location through which the plane is fixed.
• The normal vector dictates how the plane is oriented within this space.

These elements together allow us to write the equation of the plane using the format: \[ ax + by + cz = d. \] This formula describes a flat surface in three dimensions.
Understanding this concept reveals how spatial dynamics work and how structures can be described mathematically within seemingly infinite space.