Problem 38 - Thomas Calculus

Question

Find a. the direction of \(\overrightarrow{P_{1} P}_{2}\) and b. the midpoint of line segment \(P_{1} P_{2}\). \(P_{1}(0,0,0) \quad P_{2}(2,-2,-2)\)

Step-by-Step Solution

Answer

Direction: (2, -2, -2); Midpoint: (1, -1, -1).

Step 1: Identify Coordinates

Let's identify the coordinates of points \( P_1 \) and \( P_2 \). For \( P_1 \), the coordinates are \((0, 0, 0)\) and for \( P_2 \), the coordinates are \((2, -2, -2)\).

Step 2: Determine the Direction Vector

To find the direction of \( \overrightarrow{P_1P_2} \), calculate the vector by subtracting the coordinates of \( P_1 \) from \( P_2 \). This gives us: \[ \overrightarrow{P_1P_2} = (2-0, -2-0, -2-0) = (2, -2, -2). \] Thus, the direction vector is \( (2, -2, -2) \).

Step 3: Compute the Midpoint

To find the midpoint of the line segment \( P_1P_2 \), use the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right). \]Substituting the coordinates of \( P_1 \) and \( P_2 \): \[ M = \left( \frac{0+2}{2}, \frac{0-2}{2}, \frac{0-2}{2} \right) = \left( 1, -1, -1 \right). \] Therefore, the midpoint is \( (1, -1, -1) \).

Key Concepts

Direction Vector

In coordinate geometry, a direction vector helps us understand the orientation and length of a line segment between two points. Consider two points, often denoted as \(P_1\) and \(P_2\), with coordinates \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) respectively. The direction vector \(\overrightarrow{P_1P_2}\) is simply the vector created by subtracting the coordinates of \(P_1\) from those of \(P_2\).

For example, to find the direction vector from \(P_1\) to \(P_2\), you calculate:

• \(x ext{-component} = x_2 - x_1\)
• \(y ext{-component} = y_2 - y_1\)
• \(z ext{-component} = z_2 - z_1\)

This results in the direction vector \((x_2 - x_1, y_2 - y_1, z_2 - z_1)\). In our example: \(\overrightarrow{P_1P_2} = (2, -2, -2)\), revealing both the direction and the distance trend from \(P_1\) to \(P_2\).

Direction vectors are useful in many areas of geometry, including understanding the trajectory of movements, aligning objects in 3D space, or determining parallelism between lines.

Midpoint Formula

Finding the midpoint of a line segment in geometry is incredibly useful for tasks such as bisecting a segment into two equal parts or finding the center between two points. The midpoint is essentially the average of the starting and ending coordinates of a line segment.

The formula to calculate the midpoint between two points \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) is:

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

Using the points \(P_1(0, 0, 0)\) and \(P_2(2, -2, -2)\) as an example, we apply this formula:

• \(x ext{ coordinate of M} = \frac{0 + 2}{2} = 1\)
• \(y ext{ coordinate of M} = \frac{0 - 2}{2} = -1\)
• \(z ext{ coordinate of M} = \frac{0 - 2}{2} = -1\)

Thus, the midpoint \(M\) is located at \((1, -1, -1)\).

Midpoint calculations are common in various applications across mathematics, physics, and engineering, helping ensure precision in construction, mapping, and design.

3D Coordinates

In the realm of geometry, especially within three-dimensional geometry, 3D coordinates are vital for pinpointing the location of a point in space. A point in 3D is defined by an ordered triplet \((x, y, z)\), representing its position along the X, Y, and Z axes.

3D coordinates extend the concept of two-dimensional coordinates by adding depth, making them indispensable in applications such as computer graphics, physics simulations, and geographical mapping. Each dimension provides a degree of freedom, allowing for the precise positioning and manipulation of objects.

• The X-axis typically represents horizontal positioning.
• The Y-axis represents vertical positioning.
• The Z-axis adds depth, creating a space where points, lines, and shapes can exist in a simulated 3D environment.

Understanding 3D coordinates is crucial for effectively navigating this space, calculating distances, directions, midpoints, and performing transformations like rotations and translations. It all starts with grasping basic coordinate geometry concepts, such as finding direction vectors or midpoints in exercises like the one involving points \(P_1(0, 0, 0)\) and \(P_2(2, -2, -2)\).