Problem 34 - Advanced Engineering Mathematics

Question

Find the coordinates of the midpoint of the line segment between the given points. $$ (0,5,-8),(4,1,-6) $$

Step-by-Step Solution

Answer

The midpoint coordinates are (2, 3, -7).

Step 1: Understand the Midpoint Formula

The midpoint of a line segment connecting two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3-dimensional space is calculated using the formula:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right).\]

Step 2: Identify the Coordinates of Given Points

Identify the coordinates of the given points. From the problem, we have Point 1 as \((0, 5, -8)\) and Point 2 as \((4, 1, -6)\).

Step 3: Apply the Midpoint Formula

Use the midpoint formula for the coordinates \((0, 5, -8)\) and \((4, 1, -6)\): \[ \left( \frac{0 + 4}{2}, \frac{5 + 1}{2}, \frac{-8 + (-6)}{2} \right) \].

Step 4: Calculate Each Coordinate

Calculate each part of the midpoint formula:- For \(x\)-coordinate: \(\frac{0 + 4}{2} = 2\).- For \(y\)-coordinate: \(\frac{5 + 1}{2} = 3\).- For \(z\)-coordinate: \(\frac{-8 + (-6)}{2} = -7\).

Step 5: Write Down the Midpoint Coordinates

From the above calculations, the coordinates of the midpoint are \((2, 3, -7)\).

Key Concepts

3-dimensional space

In mathematics, we often deal with spaces that go beyond the two dimensions we're familiar with on a flat piece of paper. These spaces are known as 3-dimensional spaces. Here, each point is determined by three numbers, usually referred to as the coordinates. These coordinates are written in a format \((x, y, z)\), where

• \(x\) represents the position along the horizontal axis.
• \(y\) represents the position along the vertical axis.
• \(z\) represents the position along the third axis, which adds depth to our space.

Understanding 3-dimensional space is essential for visualizing objects not just on a flat surface, but in an environment similar to our real world.
It's like moving from drawing on a sheet of paper to sculpting a model—it adds an extra layer of complexity and realism. When working with 3-dimensional space, always keep in mind these three axes that define the position of every point.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the field of mathematics that uses algebraic equations to describe geometric shapes and positions. It involves using a coordinate system to simplify the math behind shapes and forms. The Cartesian coordinate system, which uses a grid of points defined by \((x, y)\) for 2D or \((x, y, z)\) for 3D, is most common.

• Points are represented as coordinates.
• Lines and curves can be expressed through equations.
• It helps in finding distances, slopes, and midpoints between points.

Coordinate geometry makes it easier to solve geometric problems by transforming them into algebraic equations. This different approach allows for precise calculations and is especially useful in fields like physics and engineering where precise measurements are required.

line segment midpoint

The midpoint of a line segment is an important concept in geometry that refers to the exact center point of a segment joining two points in space. To find this midpoint in 3-dimensional space, we use a specialized version of the midpoint formula.For any two given points, \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), the formula becomes:\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right). \]

• This formula averages out each dimension's coordinates.
• It effectively splits the line segment into two equal parts, providing a balanced center point.

Finding the midpoint is useful in navigation, construction, and any application where precise bisecting of a line segment is necessary. With this method, even in complex 3D environments, determining the center between two points becomes straightforward and concise.