Problem 38 - Advanced Engineering Mathematics

Question

Find the vector \(\overrightarrow{P_{1} P_{2}}\). $$ P_{1}(-2,4,0), P_{2}\left(6, \frac{3}{2}, 8\right) $$

Step-by-Step Solution

Answer

The vector \( \overrightarrow{P_1 P_2} \) is \((8, -\frac{5}{2}, 8)\).

Step 1: Identify Vector Coordinates

The vector \( \overrightarrow{P_1P_2} \) is the vector that goes from point \( P_1 \) to point \( P_2 \). Coordinates for \( P_1 \) are \((-2, 4, 0)\) and for \( P_2 \) are \( \left(6, \frac{3}{2}, 8\right) \).

Step 2: Apply the Vector Formula

The formula to find the vector \( \overrightarrow{P_1 P_2} \) is \( \overrightarrow{P_1 P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \). This formula calculates the difference between the coordinates of \( P_2 \) and \( P_1 \).

Step 3: Calculate the Differences for Each Component

Using the coordinates of \( P_1 \) and \( P_2 \), calculate:\[ x_2 - x_1 = 6 - (-2) = 6 + 2 = 8 \]\[ y_2 - y_1 = \frac{3}{2} - 4 = \frac{3}{2} - \frac{8}{2} = -\frac{5}{2} \]\[ z_2 - z_1 = 8 - 0 = 8 \].

Step 4: State the Resulting Vector

Thus, the vector \( \overrightarrow{P_1 P_2} \) is given by \((8, -\frac{5}{2}, 8)\).

Key Concepts

Vector Coordinates

When working with vectors, one of the first things to understand is the concept of vector coordinates. In geometry, a vector can be represented by coordinates in a coordinate system. For instance, a point in three-dimensional space, like \( P_1(-2, 4, 0) \) and \( P_2(6, \frac{3}{2}, 8) \), is expressed using three numbers that denote its position along the x, y, and z axes.
These coordinates are essential in mapping out the starting and ending points of a vector in space. Each number in the parentheses corresponds to one of the spatial dimensions:

• The first number is the x-coordinate (horizontal position).
• The second number is the y-coordinate (vertical position).
• The third number is the z-coordinate (depth position).

Knowing how to identify and work with these coordinates is the fundamental building block for operations like vector addition and subtraction.

Vector Subtraction

Vector subtraction is used to find the vector that directs from one point to another. To subtract vectors, you calculate the difference between their corresponding coordinates. This process is how we found vector \( \overrightarrow{P_1 P_2} \) in the previous example.
To compute it, use the formula:

• \(\overrightarrow{P_1 P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\)

For points \( P_1 \) and \( P_2 \), this involves calculating:

• \( x_2 - x_1 = 6 - (-2) = 8 \)
• \( y_2 - y_1 = \frac{3}{2} - 4 = -\frac{5}{2} \)
• \( z_2 - z_1 = 8 - 0 = 8 \)

Each result represents the vector's direction and magnitude in the respective coordinate direction. Understanding how to perform this operation is vital for navigating and representing vector fields.

Three-Dimensional Space

Three-dimensional space refers to a setting where every point can be described by three coordinates. It expands on the two-dimensional plane by adding a third axis, typically known as the z-axis, which provides depth.
To visualize this, imagine a cube. Each of the cube's corners can be described using three coordinates representing width (x-axis), height (y-axis), and depth (z-axis). This concept allows us to represent real-world scenarios more accurately, capturing all spatial directions.
In three-dimensional vector problems, like the calculation of \(\overrightarrow{P_1 P_2}\), each component (x, y, and z) plays an integral role. The inclusion of the z-coordinate marks a significant shift from the flat plane of two-dimensional geometry to a more comprehensive spatial understanding. Grasping how to maneuver in three-dimensional space is crucial for fields like physics, engineering, and computer graphics where spatial representation and movement must be accurately portrayed.