Problem 18 - Thomas Calculus

Question

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}.\) \(\overrightarrow{P_{1} P_{2}}\) if \(P_{1}\) is the point \((1,2,0)\) and \(P_{2}\) is the point \((-3,0,5)\)

Step-by-Step Solution

Answer

\( \overrightarrow{P_1 P_2} = -4\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} \).

Step 1: Identify the Coordinates

We start by identifying the coordinates of the given points. Point \( P_1 \) has coordinates \((1,2,0)\) and point \( P_2 \) has coordinates \((-3,0,5)\).

Step 2: Compute Components of the Vector

To find the vector \( \overrightarrow{P_1 P_2} \), calculate the difference between the coordinates of \( P_2 \) and \( P_1 \). Specifically, compute each component as follows:\v_{1} = -3 - 1 = -4\v_{2} = 0 - 2 = -2\v_{3} = 5 - 0 = 5\These give the components of vector \( \overrightarrow{P_1 P_2} \).

Step 3: Express the Vector in Vector Form

With the calculated components, express \( \overrightarrow{P_1 P_2} \) in the form \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \). Substituting the values, we get:\\( \overrightarrow{P_1 P_2} = -4\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} \).

Key Concepts

Vector Components

Vectors are mathematical entities that possess both magnitude and direction. In order to describe a vector in a three-dimensional space, we utilize its components. These components represent the vector's projection along each axis in Cartesian coordinates.
Let's break this down:

• **Components of a Vector**: Essentially, a vector component is the scalar quantity that stretches along each basis vector. In three-dimensional space, the basis vectors are typically represented by \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).
• **Finding Vector Components**: To extract the components of a vector \( \overrightarrow{P_1 P_2} \) from two points \( P_1 \) and \( P_2 \), we compute the differences of the corresponding coordinates.
• **Example**: Given the points \( P_1(1,2,0) \) and \( P_2(−3,0,5) \), the vector components are computed as \(-3 - 1 = -4\), \(0 - 2 = -2\), and \(5 - 0 = 5\).

After deriving these components, we can express the vector in its component form, \( \, -4\), \( -2\), \( 5 \), representing its reach along the X, Y, and Z axes, respectively. This understanding of vector components is vital in fields ranging from physics to computer graphics as it forms the basis for vector arithmetic.

Coordinate Geometry

Coordinate geometry—also known as analytic geometry—allows us to represent geometric figures using coordinate systems. It serves as a bridge connecting geometry and algebra through graphs and equations. This makes it easier to work with spatial objects, like vectors, using coordinates rather than purely geometric definitions.

• **Application to Vectors**: For vectors, coordinate geometry is invaluable as it allows us to express vectors as coordinate-based entities that we can manipulate algebraically.
• **Describing Lines and Vectors**: We can describe the line between two points, like \( P_1 \) and \( P_2 \), through their coordinates. This is crucial in finding vector components. The segment \( \overrightarrow{P_1 P_2} \) is inherently tied to the coordinate plane.

Coordinate geometry simplifies the process of analyzing shapes and lines, particularly in regard to locating points and determining distances or directions as vectors. By using coordinates, we transform the often complex concepts of geometry into manageable equations, enhancing our capability to visualize and calculate geometric relationships.

Cartesian Coordinates

Cartesian coordinates are the backbone of coordinate geometry. They allow us to pin down points in space using a set of numerical coordinates. Each coordinate corresponds to a distance measure along respective axes in a grid.

• **Understanding the Grid**: The Cartesian grid is formed by the intersection of perpendicular axes. Typically labeled X, Y, and Z in 3D space, each axis provides a dimension of the space.
• **Using Cartesian Coordinates in Vectors**: Vectors, like \( \overrightarrow{P_1 P_2} \), gain clarity and precision when expressed through Cartesian coordinates. Each point \( P_1 \) or \( P_2 \) is allocated three coordinates that denote its position within the space relative to these axes.
• **Working with Cartesian Vectors**: The beauty of using Cartesian coordinates lies in their versatility and alignment with fundamental algebraic operations. By subtracting the respective coordinates of two points, we can readily determine the components of a vector.

Leaning on a Cartesian plane allows for clear visualization and straightforward calculations. This makes understanding spatial relationships much easier and is crucial for solving problems involving vectors and other spatial elements, like finding directions and distances between points.