Problem 35 - Thomas Calculus

Question

In Exercises \(35-38\) , find a. the direction of \(P_{1} P_{2}\) 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

Direction vector: (3, 4, -5); Midpoint: \((\frac{1}{2}, 3, \frac{5}{2})\)."

Step 1: Calculate the Direction Vector

The direction of the line segment connecting points \(P_1(-1,1,5)\) and \(P_2(2,5,0)\) is found by determining the vector \(\vec{P_1P_2}\). The formula for this vector is given by \(\vec{P_1P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\). Substituting the coordinates in, we have \(\vec{P_1P_2} = (2 - (-1), 5 - 1, 0 - 5) = (3, 4, -5)\).

Step 2: Find the Midpoint of the Line Segment

The midpoint \(M\) of the line segment \(P_1P_2\) is given by the formula \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\). Using the coordinates \((-1, 1, 5)\) and \((2, 5, 0)\), we find the midpoint: \(M = \left( \frac{-1 + 2}{2}, \frac{1 + 5}{2}, \frac{5 + 0}{2} \right) = \left( \frac{1}{2}, 3, \frac{5}{2} \right)\).

Key Concepts

Direction Vector

A direction vector is a mathematical way to describe the orientation of a line segment in space. It's found by determining how much we "move" from one point on the line to another. For the line segment connecting two points, say \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\), the direction vector is calculated by subtracting the coordinates of \(P_1\) from \(P_2\).

Thus, the direction vector \(\vec{P_1P_2}\) is determined using:

• \(\vec{P_1P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\)

For example, if we have \(P_1(-1, 1, 5)\) and \(P_2(2, 5, 0)\), the direction vector \(\vec{P_1P_2}\) becomes \((2 - (-1), 5 - 1, 0 - 5)\), which simplifies to \((3, 4, -5)\).

This vector tells us that to move from \(P_1\) to \(P_2\), we move 3 units in the \(x\)-direction, 4 units in the \(y\)-direction, and \(-5\) units in the \(z\)-direction.

Midpoint Formula

Finding the midpoint of a line segment in 3D space is a way to find the exact center point between two endpoints. The midpoint, denoted by \(M\), is calculated by averaging corresponding coordinates of the endpoints.

The formula 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 \(P_1(-1, 1, 5)\) and \(P_2(2, 5, 0)\), we find the midpoint by substituting these values into the formula:

\(M = \left( \frac{-1 + 2}{2}, \frac{1 + 5}{2}, \frac{5 + 0}{2} \right) = \left( \frac{1}{2}, 3, \frac{5}{2} \right)\).

This means the midpoint \(M\) of the line segment is \(\left( \frac{1}{2}, 3, \frac{5}{2} \right)\), effectively splitting the line segment into two equal halves.

3D Coordinate Geometry

3D Coordinate Geometry is a branch of mathematics that deals with points, lines, and other figures in three-dimensional space. It extends the principles of 2D coordinate geometry by adding a third dimension, allowing for more complex analysis and calculations.

In 3D space, a point is represented by an ordered triplet, \((x, y, z)\). Here:

• \(x\) is the horizontal position
• \(y\) is the depth (when viewing front-on)
• \(z\) is the vertical position

To better understand lines in 3D:

• Direction vectors help to express the orientation in space.
• The midpoint formula finds the center between two points.

These principles allow mathematicians and engineers to perform complex spatial analyses, solve problems involving spatial distances, and model physical phenomena accurately.