Problem 35 - Thomas Calculus

Question

Find a. the direction of \(\overrightarrow{P_{1} P}_{2}\) and b. the midpoint of line segment \(P_{1} P_{2}\). \(P_{1}(-1,1,5) \quad P_{2}(2,5,0)\)

Step-by-Step Solution

Answer

The direction is \( (3, 4, -5) \) and the midpoint is \( \left( \frac{1}{2}, 3, \frac{5}{2} \right) \).

Step 1: Identify Component Differences

To find the direction of \( \overrightarrow{P_{1}P_{2}} \), calculate the differences in the corresponding components of \( P_1 \) and \( P_2 \). The direction vector components can be found by subtracting the coordinates of \( P_1 \) from the coordinates of \( P_2 \). Let \( P_1(x_1, y_1, z_1) = (-1, 1, 5) \) and \( P_2(x_2, y_2, z_2) = (2, 5, 0) \). Thus, the direction vector \( \overrightarrow{P_1 P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (2 - (-1), 5 - 1, 0 - 5) = (3, 4, -5) \).

Step 2: Result of Direction

The direction of \( \overrightarrow{P_1 P_2} \) is given by the vector \( (3, 4, -5) \). This provides the relative direction from point \( P_1 \) to point \( P_2 \) in three-dimensional space.

Step 3: Calculate Midpoint Formula

To find the midpoint of line segment \( P_1 P_2 \), use the midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \). Substitute the coordinates: \( M = \left( \frac{-1 + 2}{2}, \frac{1 + 5}{2}, \frac{5 + 0}{2} \right) \). Simplifying, we find: \( M = \left( \frac{1}{2}, 3, \frac{5}{2} \right) \).

Step 4: Result of Midpoint

The midpoint of the line segment \( P_1 P_2 \) is \( \left( \frac{1}{2}, 3, \frac{5}{2} \right) \). This is the center point between \( P_1 \) and \( P_2 \) in three-dimensional space.

Key Concepts

Understanding Direction Vectors

A direction vector is a mathematical entity that gives us the direction from one point to another. In the context of the exercise, we have points \( P_1(-1, 1, 5) \) and \( P_2(2, 5, 0) \) in three-dimensional space. The direction vector \( \overrightarrow{P_1 P_2} \) essentially tells us how to move from \( P_1 \) to \( P_2 \) directly.

To calculate this, we subtract each coordinate of \( P_1 \) from \( P_2 \):

• \( x \)-component: \( 2 - (-1) = 3 \)
• \( y \)-component: \( 5 - 1 = 4 \)
• \( z \)-component: \( 0 - 5 = -5 \)

Putting these together, the direction vector is \( (3, 4, -5) \). This vector tells us to move 3 units in the \( x \)-direction, 4 units in the \( y \)-direction and -5 units in the \( z \)-direction. This directional information is crucial when navigating or solving problems in space.

Demystifying the Midpoint Formula

The midpoint formula is a simple yet powerful tool for finding the center point between two locations in space. It gives us a point that is exactly halfway between our two points of interest. Let's say you are given points \( P_1(-1, 1, 5) \) and \( P_2(2, 5, 0) \).

To find the midpoint, we need to average out the coordinates individually:

• \( x \)-coordinate: \( \frac{-1 + 2}{2} = \frac{1}{2} \)
• \( y \)-coordinate: \( \frac{1 + 5}{2} = 3 \)
• \( z \)-coordinate: \( \frac{5 + 0}{2} = \frac{5}{2} \)

Hence, the midpoint is \( \left( \frac{1}{2}, 3, \frac{5}{2} \right) \). This point is incredibly significant as it is exactly equidistant from both \( P_1 \) and \( P_2 \), providing a perfect balance in the three-dimensional space.

Navigating Three-Dimensional Space

Three-dimensional space is a fascinating realm where we deal with points having coordinates \( (x, y, z) \). Unlike two-dimensional space, which only has length and width, the third dimension adds depth, providing a more comprehensive way to describe locations.

When working with vectors and points in this space, it's crucial to consider all three dimensions:

• X-axis: Refers to the horizontal direction.
• Y-axis: Denotes the vertical direction.
• Z-axis: Represents depth, giving the sense of height above and below the xy-plane.

Vectors in this space, such as the direction vector \( \overrightarrow{P_1 P_2} \), help map paths between points.
Understanding this spatial orientation enables us to solve problems accurately and visualize the placement of objects in a more relatable manner.