Problem 37 - University Calculus: Early Transcendentals

Question

Find a. the direction of \(\vec{P}_{1} \vec{P}_{2}\) and \(\mathbf{b}\). the midpoint of line segment \(P_{1} P_{2}\). $$P_{1}(3,4,5) \quad P_{2}(2,3,4)$$

Step-by-Step Solution

Answer

Direction: \((-1, -1, -1)\), Midpoint: \(\left(\frac{5}{2}, \frac{7}{2}, \frac{9}{2}\right)\)."

Step 1: Understand the problem

Given two points, \(P_1(3,4,5)\) and \(P_2(2,3,4)\), we need to find the direction vector and the midpoint of the line segment connecting these two points.

Step 2: Find the Direction Vector

The direction vector \(\vec{P}_1 \vec{P}_2\) from point \(P_1\) to point \(P_2\) can be found by subtracting the coordinates of \(P_1\) from \(P_2\). So, \(\vec{P}_1 \vec{P}_2 = (2-3, 3-4, 4-5) = (-1, -1, -1)\).

Step 3: Calculate the Midpoint

The midpoint \(M\) of a line segment with endpoints \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by the formula \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)\). Applying this to our points, the midpoint \(M\) is \(\left(\frac{3+2}{2}, \frac{4+3}{2}, \frac{5+4}{2}\right) = \left(\frac{5}{2}, \frac{7}{2}, \frac{9}{2}\right)\).

Key Concepts

Midpoint Formula

The midpoint formula is a mathematical tool that helps us find the center point of a line segment connecting two given points in space. In simple terms, the midpoint is the average of the x, y, and z coordinates of the endpoints in 3D space.

To calculate the midpoint \(M\) of a line segment whose endpoints are \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), use the formula:

• \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\)

For instance, if you have two points, \(P_1(3, 4, 5)\) and \(P_2(2, 3, 4)\), simply plug these into the formula. This gives you:

• Midpoint \(M = \left( \frac{3 + 2}{2}, \frac{4 + 3}{2}, \frac{5 + 4}{2} \right) = \left( \frac{5}{2}, \frac{7}{2}, \frac{9}{2} \right) \)

This point \(\left( \frac{5}{2}, \frac{7}{2}, \frac{9}{2} \right)\) represents the exact center of the line segment joining \(P_1\) and \(P_2\). This method is useful when you need to find a central point which could be crucial in 3D graphics, architecture, or any field involving spatial calculations.

3D Coordinates

3D coordinates are like addresses on a three-dimensional map, specifying precise locations in space. Each point in 3D space is represented by three values: \((x, y, z)\).

• The 'x' coordinate tells how far the point is along the horizontal axis.
• The 'y' coordinate indicates the vertical position.
• The 'z' coordinate represents depth or how far out from the 2D base plane the point is.

In our problem, points \(P_1(3, 4, 5)\) and \(P_2(2, 3, 4)\) are used to describe locations in 3D space.

Understanding 3D coordinates is crucial in many fields such as physics, computer graphics, and robotics, where precise positioning in space is required. With 3D coordinates, one can model objects, simulate environments, or even navigate spaces in virtual reality.

Line Segment

A line segment in mathematics is the part of a line that connects two distinct points, having both definite endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a fixed length.

The points \(P_1\) and \(P_2\) in our exercise represent the endpoints of the line segment. Calculating properties such as the midpoint or direction vector of a line segment helps better understand its characteristics and relations in space.

Key properties you'll often calculate for a line segment include:

• Midpoint: Using the midpoint formula to find the center of the segment.
• Direction Vector: To understand its orientation from one point to the other.
• Length: Calculated using the distance formula to know how long the segment is.

Line segments are fundamental in geometry and are instrumental in construction, design, and computer modeling. These simple constructs form the basis of complex shapes and structures, making understanding their properties essential.