Problem 36 - Advanced Engineering Mathematics

Question

Let \(P_{3}\) be the midpoint of the line segment between \(P_{1}(-3,4,1)\) and \(P_{2}(-5,8,3)\). Find the coordinates of the midpoint of the line segment (a) between \(P_{1}\) and \(P_{3}\) and (b) between \(P_{3}\) and \(P_{2}\).

Step-by-Step Solution

Answer

(a) (-3.5, 5, 1.5); (b) (-4.5, 7, 2.5).

Step 1: Calculate the Coordinates of Midpoint P3

The midpoint formula in three dimensions between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by \[ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right) \].For points \(P_1(-3,4,1)\) and \(P_2(-5,8,3)\), apply the formula:- For the x-coordinate: \(\frac{-3 + (-5)}{2} = \frac{-8}{2} = -4\)- For the y-coordinate: \(\frac{4 + 8}{2} = \frac{12}{2} = 6\)- For the z-coordinate: \(\frac{1 + 3}{2} = \frac{4}{2} = 2\)Thus, \(P_3\) has coordinates \((-4, 6, 2)\).

Step 2: Find the Midpoint Between P1 and P3

Using the midpoint formula on points \(P_1(-3,4,1)\) and \(P_3(-4,6,2)\):- For the x-coordinate: \(\frac{-3 + (-4)}{2} = \frac{-7}{2} = -3.5\)- For the y-coordinate: \(\frac{4 + 6}{2} = \frac{10}{2} = 5\)- For the z-coordinate: \(\frac{1 + 2}{2} = \frac{3}{2} = 1.5\)Thus, the midpoint of the line segment between \(P_1\) and \(P_3\) is \((-3.5, 5, 1.5)\).

Step 3: Find the Midpoint Between P3 and P2

Using the midpoint formula on points \(P_3(-4,6,2)\) and \(P_2(-5,8,3)\):- For the x-coordinate: \(\frac{-4 + (-5)}{2} = \frac{-9}{2} = -4.5\)- For the y-coordinate: \(\frac{6 + 8}{2} = \frac{14}{2} = 7\)- For the z-coordinate: \(\frac{2 + 3}{2} = \frac{5}{2} = 2.5\)Thus, the midpoint of the line segment between \(P_3\) and \(P_2\) is \((-4.5, 7, 2.5)\).

Key Concepts

Three-Dimensional Geometry

Three-dimensional geometry is an extension of the familiar two-dimensional plane into an extra dimension. Instead of dealing with just length and width, we include height. This introduces a third axis, the z-axis, in addition to the x and y-axes.
In our problem, we are dealing with points in three-dimensional space, which are represented by coordinate triples \(x, y, z\). This allows us to visualize objects that have depth and exist in a 3D realm, mirroring real-world physical structures.
Understanding 3D geometry is fundamental for fields like engineering, architecture, and computer graphics. It enables the modeling of complex shapes and the calculation of distances, angles, and volumes in space.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves placing geometric figures in a coordinate plane and finding properties such as lengths, midpoints, and angles using numerical equations.
It helps us describe and analyze geometric shapes purely through mathematics rather than relying solely on a visual perspective.
In the context of our exercise, coordinate geometry provides tools like the vector forms of points, and the formulae we use to find the midpoint between two points.
Using the midpoint formula \[ \( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \) \], we can identify the precise center point between any two given points in a 3D space.

Mathematics Problem Solving

Effective problem solving in mathematics involves understanding the problem, devising a plan, carrying out the plan, and then examining the solution for accuracy.
This classic approach can be applied to our exercise: we first need to comprehend what finding a midpoint entails and what the inputs are.
Next, we apply the midpoint formula in the context of three dimensions, using our known points.

• Understand known points: \((-3,4,1)\) and \((-5,8,3)\).
• Calculate intermediate midpoint: \((-4,6,2)\).
• Use results to find additional midpoints.

By systematically working through the problem, step-by-step, we ensure that we fully understand the maths and minimize mistakes.
This process is applicable to a wide range of mathematical challenges, building confidence in navigating more complex questions.