Problem 37 - Advanced Engineering Mathematics

Question

Find the vector \(\overrightarrow{P_{1} P_{2}}\). $$ P_{1}(3,4,5), P_{2}(0,-2,6) $$

Step-by-Step Solution

Answer

The vector \( \overrightarrow{P_{1} P_{2}} \) is \((-3, -6, 1)\).

Step 1: Understand Vector Notation

Vectors are often represented as directed line segments. We describe a vector between two points in space by subtracting coordinates.

Step 2: Apply the Formula for Vector Subtraction

To find the vector \( \overrightarrow{P_{1}P_{2}} \), we apply the formula \( \overrightarrow{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \), where \( A = (x_1, y_1, z_1) \) and \( B = (x_2, y_2, z_2) \).

Step 3: Substitute the Coordinates of Points

Given points are \( P_1(3,4,5) \) and \( P_2(0,-2,6) \). Substitute them: \( \overrightarrow{P_{1}P_{2}} = (0 - 3, -2 - 4, 6 - 5) \).

Step 4: Perform the Arithmetic Operations

Calculate each component: \( 0 - 3 = -3 \), \( -2 - 4 = -6 \), and \( 6 - 5 = 1 \).

Step 5: Write the Resultant Vector

Combine the calculated components to write the resultant vector: \( \overrightarrow{P_{1}P_{2}} = (-3, -6, 1) \).

Key Concepts

Vector Notation

Vector notation is a way of representing mathematical quantities that have both a direction and magnitude. In simple terms, vectors typically appear as arrow-tipped line segments in diagrams. In the context of coordinates, vectors are expressed as ordered tuples or lists of numbers that correspond to their components in a particular coordinate system. For instance, a three-dimensional vector can be written as

• \((x, y, z)\)

where each number represents the vector's component in the corresponding axis direction.
Understanding vector notation is crucial when dealing with vector subtraction, as it provides a clear and structured way to express the difference between two points. This makes it easier to calculate and visualize what the vector represents.When vectors are expressed between two points, say from point \( P_1 \) to point \( P_2 \), the notation \( \overrightarrow{P_{1}P_{2}} \) signifies the vector pointing from the first point to the second.

Coordinate System

A coordinate system is a framework for defining and specifying the positions of points in space. The typical coordinate systems used in vector mathematics include

• Cartesian
• Polar
• Spherical

The Cartesian coordinate system is the most common, where each point in space is described by a set of numbers.
For example, in a three-dimensional space, each point has coordinates \((x, y, z)\). These coordinates tell you the point's location along the x-axis, y-axis, and z-axis respectively.
In our exercise, the points \( P_1(3,4,5) \) and \( P_2(0,-2,6) \) are in a three-dimensional Cartesian coordinate system.
Each point is represented by its x, y, and z components, making arithmetic operations, like subtraction, straightforward.

Arithmetic Operations

Arithmetic operations allow us to manipulate numbers and quantities, which is essential for calculations involving vectors. With vectors, the most common operation is subtraction to find the vector between two points. This involves subtracting the coordinates of one point from another.
To find the vector \( \overrightarrow{P_{1}P_{2}} \), we apply the formula:

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

This formula helps to calculate each component of the resulting vector.
From the given points \( P_1(3,4,5) \) and \( P_2(0,-2,6) \), we subtract their coordinates:

• \((0 - 3, -2 - 4, 6 - 5)\)

Performing these operations gives us:

• \(-3\) for the x-component
• \(-6\) for the y-component
• \(1\) for the z-component

Together, they form the resultant vector: \( \overrightarrow{P_{1}P_{2}} = (-3, -6, 1) \), showing the direction and magnitude from \( P_1 \) to \( P_2 \).