Problem 32 - University Calculus: Early Transcendentals

Question

Find the distance from the point (-2,1,4) to the a. plane \(x=3\) b. plane \(y=-5\) c. plane \(z=-1\)

Step-by-Step Solution

Answer

a. 5; b. 6; c. 5.

Step 1: Understanding the distance to the x-plane

The problem asks us to find the distance from a point \((-2,1,4)\) to the plane \(x=3\). Planes parallel to the yz-plane have a constant x-coordinate. The distance from the point (-2,1,4) to the plane \(x=3\) is simply the absolute difference in the x-coordinates of the point and the plane, because they lie along a line parallel to the x-axis.

Step 2: Calculating the x-distance

To find the distance from the point \((-2, 1, 4)\) to the plane \(x=3\), subtract the x-coordinate of the point from the x-value of the plane and take the absolute value. \(\text{Distance}= |3 - (-2)| = |3 + 2| = 5.\)

Step 3: Understanding the distance to the y-plane

The plane \(y=-5\) is parallel to the xz-plane and has a constant y-coordinate. Similarly, the distance from the point \((-2,1,4)\) to the plane \(y=-5\) depends only on the difference in y-coordinates.

Step 4: Calculating the y-distance

To find the distance from the point \((-2, 1, 4)\) to the plane \(y=-5\), subtract the y-coordinate of the point from the y-value of the plane and take the absolute value: \(\text{Distance}= |-5 - 1| = |-6| = 6.\)

Step 5: Understanding the distance to the z-plane

The plane \(z=-1\) is parallel to the xy-plane and has a constant z-coordinate. The z-distance is the difference between the z-coordinate of the point and the z-coordinate of the plane.

Step 6: Calculating the z-distance

To find the distance from the point \((-2, 1, 4)\) to the plane \(z=-1\), subtract the z-coordinate of the point from the z-value of the plane and take the absolute value: \(\text{Distance}= |-1 - 4| = |-5| = 5.\)

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as Cartesian geometry, is a fundamental branch of mathematics where geometric problems are solved using a coordinate system. This system utilizes coordinates on planes to define points and lines. It plays a crucial role when we want to represent points like \((-2, 1, 4)\) in a three-dimensional space.

• Coordinates: A point in 3D space is defined by three numbers \((x, y, z)\), each representing a distance along the corresponding axis on the Cartesian plane.
• Distance Calculations: To find the distance between a point and a plane, you mainly identify differences in one coordinate, while the other two remain constant.

This approach makes problems in three-dimensional space much more manageable.In this exercise, coordinate geometry is applied by calculating simple distances from a given point to planes characterized by constant x, y, or z values. These concepts are straightforward in coordinate geometry, as we can subtract the constant value of the plane from the coordinate of the point, and take its absolute value to find the precise perpendicular distance.

3D Geometry

In 3D geometry, we explore shapes and constructs in space defined by three dimensions: length, width, and height. Each point in this space is designated by three coordinates. Three-dimensional geometry allows us to perform an array of operations, crucial for both theoretical mathematics and practical applications.

• Visual Representation: Visualizing points, lines, and planes becomes significant when dealing with three-dimensional space.
• Distances: Unlike 2D, where distances are limited to horizontal or vertical, in 3D, distances account for depth alongside traditional length and width.

For this specific task, 3D geometry simplifies our distance-finding operation. By only dealing with changes along the respective axis (i.e., x-axis, y-axis, or z-axis), calculations are much more manageable without delving into complex formulas.

Plane Equations

A plane in mathematics is a flat, two-dimensional surface extending infinitely in 3D space. To fully understand the problem involving calculating the distance from a point to a plane, one must grasp plane equations and properties. In our instance, planes are showcased with equations like \(x=3\), \(y=-5\), and \(z=-1\). Knowing these will help determine their orientation and behavior in space.

• Standard forms: When a plane is written as \(x = c\), it represents all points for any y and z, but with a fixed x-value.
• Parallelism: Planes are always parallel to the axis that isn't in their equation. For instance, the plane \(x=3\) is parallel to the yz-plane.

Understanding these concepts makes the task of finding a perpendicular distance less challenging.With plane equations, you can quickly determine positions and distances in 3D by applying simple subtraction and absolute values in the given dimension.