Problem 19 - APEX Calculus

Question

Give the equation of the described plane in standard and general forms. Contains the point (-4,7,2) and is parallel to the plane \(3(x-2)+8(y+1)-10 z=0\)

Step-by-Step Solution

Answer

The plane equation is: standard form \(3x + 8y - 10z - 24 = 0\), general form \(3x + 8y - 10z = 24\).

Step 1: Identify the properties of the given plane

The provided plane is given as \(3(x-2) + 8(y+1) - 10z = 0\). This can be simplified to identify plane properties: the normal vector to this plane is \((3, 8, -10)\). The plane we need is parallel to this one, therefore it shares the same normal vector.

Step 2: Use the point-normal form of a plane equation

To find the equation of a plane that is parallel to a given plane and contains a specific point, we can use the point-normal form of a plane equation: \(a(x-x_0) + b(y-y_0) + c(z-z_0) = 0\). Here, \((a, b, c)\) is the normal vector, and \((x_0, y_0, z_0)\) is the point on the plane.

Step 3: Plug in the known values

We have a normal vector \((3, 8, -10)\) and a point \((-4, 7, 2)\). Plug these values into the point-normal form equation: \[3(x + 4) + 8(y - 7) - 10(z - 2) = 0\].

Step 4: Simplify to find the standard form

Expand and simplify the equation: \[3(x + 4) + 8(y - 7) - 10(z - 2) = 0\] simplifies to \[3x + 12 + 8y - 56 - 10z + 20 = 0\], which further simplifies to \[3x + 8y - 10z - 24 = 0\]. This is the standard form of the plane equation.

Step 5: Consider the general form

For the general form of the plane equation, we simply rearrange the terms to make sure all terms are on one side equal to zero, with the constant on the right-hand side if needed. Thus, it matches the format found in the previous step: \[3x + 8y - 10z = 24\].

Key Concepts

Normal Vector

The normal vector is an essential component when working with planes in three-dimensional space. It serves as a directional guide indicating how the plane is oriented in space. Elements of the normal vector are perpendicular to the plane surface.
Understanding the normal vector simplifies many geometrical problems related to planes.

• The normal vector is typically represented as \( \mathbf{n} = (a, b, c) \).
• These values \(a, b,\) and \(c\) are taken directly from the plane equation coefficients in the form \(ax + by + cz = d\).
• Any plane parallel to another shares the same normal vector.

By identifying the normal vector from a given plane equation, you can easily determine the orientation of other parallel planes.

Point-Normal Form

The point-normal form is a valuable equation configuration. It helps us conveniently express the equation of a plane when a point on the plane and a normal vector are known.
This form is expressed as:

• \(a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\)
• Here, \((x_0, y_0, z_0)\) is a given point on the plane.
• \((a, b, c)\) are the components of the normal vector.

To effectively utilize this form, replace the variables with the known point and the normal vector, making it straightforward to derive the equation of a plane given these properties.

Parallel Planes

Parallel planes are two or more planes in three-dimensional space that never intersect. This is possible only if they share the same normal vector but have different constant terms.
Here's what you need to know about parallel planes:

• Same normal vector \((a, b, c)\) implies parallelism.
• A shift in the constant term \(d\) in the general form changes the position of the plane without affecting its orientation.
• Identical sets of coefficients before \(x, y,\) and \(z\) mean two planes are parallel.

Their fixed orientation yet distinct position makes parallel planes an intriguing feature of spatial geometry.

Standard Form of a Plane

The standard form of a plane equation is a simplified way to depict a plane. It is particularly useful for systematic problemsolving and recognition of plane properties.
Standard form of a plane takes on:

• \(ax + by + cz = d\)
• Where \(a, b, c\) are components of the normal vector.
• \(d\) is a constant representing its position in space relative to the origin.

Converting a plane to its standard form involves expanding and simplifying the point-normal form equation and arranging terms accordingly.

General Form of a Plane

The general form of a plane is simply a rearrangement of the standard form, often used interchangeably with it. It highlights the relationship between variables and the plane's position in three-dimensional space.

• This form still utilizes: \(ax + by + cz = d\)
• Reiterates the plane's equation with emphasis on equilibrium among terms.
• Simplifies visualizing and determining key relationships, like parallelism, between planes.

The general form incorporates all crucial components in a flexible yet concise manner, making it a staple in various analytical and geometric considerations.