Problem 36 - Thomas Calculus

Question

Find a. the direction of \(\overrightarrow{P_{1} P}_{2}\) and b. the midpoint of line segment \(P_{1} P_{2}\). \(P_{1}(1,4,5) \quad P_{2}(4,-2,7)\)

Step-by-Step Solution

Answer

Direction: \((3, -6, 2)\); Midpoint: \(\left(\frac{5}{2}, 1, 6\right)\)."

Step 1: Find the Direction Vector

To find the direction vector of \( \overrightarrow{P_{1}P_{2}} \), subtract the coordinates of \( P_{1} \) from the coordinates of \( P_{2} \). This gives: \[ \overrightarrow{P_{1}P_{2}} = (4 - 1, -2 - 4, 7 - 5) = (3, -6, 2) \] Thus, the direction vector is \( \overrightarrow{d} = (3, -6, 2) \).

Step 2: Calculate the Midpoint

The midpoint of a line segment \( \overline{P_{1}P_{2}} \) is found by taking the average of the corresponding coordinates of \( P_{1} \) and \( P_{2} \). Calculate each coordinate:\[ x = \frac{1+4}{2} = \frac{5}{2} \]\[ y = \frac{4+(-2)}{2} = \frac{2}{2} = 1 \]\[ z = \frac{5+7}{2} = \frac{12}{2} = 6 \]Thus, the midpoint is \( M = \left( \frac{5}{2}, 1, 6 \right) \).

Key Concepts

Direction Vector

The concept of a direction vector is crucial in vector mathematics. In simple terms, it's a vector that points from one point to another, providing both direction and rate of movement. Imagine standing at point \( P_1 \) and wanting to walk to point \( P_2 \). The direction vector tells you:

• Which way to go (direction)
• How far each step should be in each dimension (rate of change)

To find the direction vector between two points, you subtract the coordinates of the starting point from the ending point. For this particular exercise, our points are \( P_1(1, 4, 5) \) and \( P_2(4, -2, 7) \).
By subtracting, we find:
\[\overrightarrow{P_{1}P_{2}} = (4 - 1, -2 - 4, 7 - 5) = (3, -6, 2)\]
This results in the direction vector \( \overrightarrow{d} = (3, -6, 2) \). It tells us that the point moves 3 units in the x-direction, -6 units in the y-direction, and 2 units in the z-direction from \( P_1 \) to \( P_2 \).
Understanding direction vectors allows for navigation in space, modeled easily in three-dimensional coordinate geometry.

Midpoint Formula

The midpoint of a line segment in coordinate geometry is the exact middle point that divides the segment into two equal parts. Think of it as the average position between two given points in a space. Calculating this involves a simple average for each coordinate dimension: x, y, and z.
For points \( P_1(1, 4, 5) \) and \( P_2(4, -2, 7) \), the midpoint \( M \) is found by averaging corresponding coordinates:

• For the x-coordinate: \( \frac{1+4}{2} = \frac{5}{2} \)
• For the y-coordinate: \( \frac{4 + (-2)}{2} = 1 \)
• For the z-coordinate: \( \frac{5+7}{2} = 6 \)

So, the midpoint \( M \) becomes \( \left( \frac{5}{2}, 1, 6 \right) \).
This midpoint calculation is fundamental in various applications such as finding the center of gravity, navigation, and more, ensuring symmetry and balance in geometric shapes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic techniques to solve geometric problems on a coordinate plane. It bridges the gap between algebra and geometry, allowing us to determine distances, angles, and other geometric properties using equations and formulas.
The primary tools of coordinate geometry include the distance formula, slope, and midpoint formula, which we utilized in our exercise. It extends to three dimensions by adding a z-coordinate, thus enabling the study of points, lines, and planes that exist in space, beyond just a flat surface.

• **Direction Vectors** determine how to move from one point to another.
• **Midpoints** find the center between two points.
• **Equations** describe lines and curves mathematically.

Coordinate geometry facilitates precise calculations and modeling in various fields, such as physics, engineering, and computer graphics, making it an indispensable tool for professionals and students alike.