Problem 36 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The direction is \( \langle 3, -6, 2 \rangle \) and the midpoint is \( \left( \frac{5}{2}, 1, 6 \right) \).

Step 1: Determine the Vector Direction

To find the direction of the vector \( \vec{P}_{1} \vec{P}_{2} \), we calculate it as the difference between point \( P_2 \) and point \( P_1 \). The formula to find the vector direction is:\[\vec{P}_{1} \vec{P}_{2} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle\]Plug in the coordinates of \( P_{1}(1,4,5) \) and \( P_{2}(4,-2,7) \):\[\vec{P}_{1} \vec{P}_{2} = \langle 4 - 1, -2 - 4, 7 - 5 \rangle = \langle 3, -6, 2 \rangle\]Thus, the direction of \( \vec{P}_{1} \vec{P}_{2} \) is \( \langle 3, -6, 2 \rangle \).

Step 2: Calculate the Midpoint

The midpoint \( M \) of a line segment between two points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \) is given by the formula:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\]Using the coordinates of \( P_{1}(1,4,5) \) and \( P_{2}(4,-2,7) \), the midpoint is calculated as follows:\[M = \left( \frac{1 + 4}{2}, \frac{4 + (-2)}{2}, \frac{5 + 7}{2} \right) = \left( \frac{5}{2}, 1, 6 \right)\]Therefore, the midpoint of the line segment \( P_1P_2 \) is \( \left( \frac{5}{2}, 1, 6 \right) \).

Key Concepts

Vector Direction

In the world of vector mathematics, understanding vector direction is crucial. A vector has both magnitude and direction, which allows it to represent anything from a moving object's velocity to forces acting in physical systems. The direction of a vector essentially tells us where it is pointing. When we compute the direction of a vector joining two points, we are concerned with the difference in coordinates that define a straight path from the initial point to the terminal point.To find the direction of vector \( \vec{P}_{1} \vec{P}_{2} \) originating from point \( P_1 \) to point \( P_2 \), we subtract the corresponding coordinates of \( P_1 \) from \( P_2 \):

• Subtract the \( x \) coordinate of \( P_1 \) from the \( x \) coordinate of \( P_2 \)
• Subtract the \( y \) coordinate of \( P_1 \) from the \( y \) coordinate of \( P_2 \)
• Subtract the \( z \) coordinate of \( P_1 \) from the \( z \) coordinate of \( P_2 \)

Given \( P_1(1,4,5) \) and \( P_2(4,-2,7) \), the resulting vector \( \vec{P}_{1} \vec{P}_{2} \) is calculated as:\[ \vec{P}_{1} \vec{P}_{2} = \langle 4 - 1, -2 - 4, 7 - 5 \rangle = \langle 3, -6, 2 \rangle \]This tells us that the vector moves 3 units in the positive \( x \)-direction, 6 units in the negative \( y \)-direction, and 2 units in the positive \( z \)-direction.

Midpoint of a Line Segment

The midpoint of a line segment in geometry is an average point that divides the segment into two equal parts.Calculating the midpoint is a straightforward yet powerful tool in both two-dimensional and three-dimensional geometry.It's the geographical center between two endpoints of a line segment.To find the midpoint \( M \) between two points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \), use the formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]Inserting our points \( P_1(1,4,5) \) and \( P_2(4,-2,7) \) into this formula results in:\[ M = \left( \frac{1 + 4}{2}, \frac{4 + (-2)}{2}, \frac{5 + 7}{2} \right) = \left( \frac{5}{2}, 1, 6 \right) \]This midpoint tells us that if one were to travel from \( P_1 \) to \( P_2 \), they would be equally halfway at point \( \left( \frac{5}{2}, 1, 6 \right) \). This principle can also be used to find central points within data sets or in creating balances and checks within coordinates.

Three-Dimensional Geometry

Three-dimensional geometry allows us to visualize and solve problems that involve depth in addition to the height and width managed in two dimensions.This geometry deals with points plotted in space using three coordinates (\( x, y, z \)), allowing us to model scenarios in the real world where every object and movement is three-dimensional.A key element of three-dimensional geometry is the Cartesian coordinate system, where every point in space is defined by a tuple \((x, y, z)\):

• \( x \): the horizontal position
• \( y \): the vertical position
• \( z \): the depth position

Using three-dimensional geometry, we find the vector directions and midpoints as discussed in our previous sections. This gives us clear and precise measurements of how one point in space relates to another. Such tools are essential in fields ranging from computer graphics to physics and engineering, where understanding spatial qualities is fundamental.Mastering three-dimensional geometry enhances analytical and spatial reasoning skills, broadening one's capability in many scientific and practical applications.