Problem 17 - Calculus an Applied Approach

Question

In Exercises \(17-20,\) find the coordinates of the midpoint of the line segment joining the two points. $$ (6,-9,1),(-2,-1,5) $$

Step-by-Step Solution

Answer

The coordinates of the midpoint of the line segment joining the two points (6,-9,1) and (-2,-1,5) is (2, -5, 3).

Step 1: Identify Coordinates

Identify the coordinates of the two points. Call the first point (6,-9,1) as A and the second point (-2,-1,5) as B.

Step 2: Apply Midpoint Formula

Find the midpoint by applying the midpoint formula, \([ (x_1+x_2)/2, (y_1+y_2)/2 ,(z_1+z_2)/2 ]\), to find the coordinates of the midpoint M, which are \([ (6+(-2))/2, ((-9)+(-1))/2 ,(1+5)/2 ]\).

Step 3: Calculate the Midpoint

Simplify the coordinates of the midpoint M to get \([ (6-2)/2, ((-9-1))/2 ,(1+5)/2 ]\) which simplifies further to \([ 4/2, -10/2 , 6/2 ]\). Simplify further to obtain the calculated midpoint as (2, -5, 3).

Key Concepts

Coordinates

In geometry, coordinates serve as a way of finding or specifying the position of points in space. When dealing with any kind of geometry, knowing how to find and use coordinates is crucial. Let's look at how coordinates function in three-dimensional space. Imagine a traditional graph you've seen in school. It uses two axes, the x-axis and the y-axis. This is called a two-dimensional coordinate system because it can describe points in a flat plane. However, in three-dimensional geometry, there's an additional z-axis, allowing us to describe locations in space, where each point is represented as (x, y, z).

• The x-coordinate indicates the point’s position along the horizontal axis.
• The y-coordinate shows the point’s position along the vertical plane.
• The z-coordinate reveals the point’s height or depth in space.

Understanding how to plot and interpret these coordinates is vital for solving problems concerning the spatial position of objects, like finding the midpoint of a line segment in 3D geometry.

Midpoint formula

The midpoint formula is a useful tool in geometry, especially when you want to find the point that's exactly halfway between two other points in 3D space. This formula takes the average of the x, y, and z coordinates of the given points. When faced with two points, say Point A: (x₁, y₁, z₁)
Point B: (x₂, y₂, z₂),
the formula for the midpoint M is:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\]This formula helps in multiple scenarios, such as ensuring symmetry in designs or solving geometric problems. Importantly, by finding the midpoint, you are effectively dividing the line segment into two equal parts. This skill is valuable across various fields, including computer graphics, architecture, and engineering. Use the formula whenever you need precision and accuracy in locating the midpoint of any line segment between two points in space.

Three-dimensional geometry

Three-dimensional geometry takes geometry from flat surfaces into the world of depth, providing a complete framework to analyze spaces and forms that possess length, breadth, and height. In 3D geometry, calculations and visualizations allow you to understand space in more realistic terms. A three-dimensional coordinate system is made up of three axes:

• X-axis: Usually the horizontal line, used for width.
• Y-axis: Typically the vertical line, indicating height.
• Z-axis: Represents depth, coming in and out of view.

Each point in 3D space is expressed with three numbers (x, y, z), which shows its position relative to these axes. Three-dimensional geometry embraces concepts like vectors, planes, and solids, which extend beyond the flat criteria of 2D geometry. This level of geometry helps in understanding and designing three-dimensional objects and spaces accurately. For example, when calculating the midpoint in a line segment, understanding 3-dimensional geometry allows us to precisely locate points in a space that an object lies within. This knowledge will aid in fields such as robotics, virtual reality, and physics, where a grasp of space and form is essential.