Problem 92 - Algebra and Trigonometry

Question

Explaining the Concepts. What is the vector \(\mathbf{j} ?\)

Step-by-Step Solution

Answer

The vector \( \mathbf{j} \) is a unit vector in the positive y direction in a three-dimensional Cartesian coordinate system. It is used to show the orientation or direction of a vector in the y-axis direction, and to represent a vector in three-dimensional space.

Step 1: Definition of Vector \( \mathbf{j} \)

In three-dimensional Cartesian coordinate system, vectors are often expressed as the sum of three components: \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). Each of these components represent the unit vector along the x, y, and z-axes respectively. Specifically, \( \mathbf{j} \) is the unit vector along the y-axis.

Step 2: Usage of Vector \( \mathbf{j} \)

The unit vector \( \mathbf{j} \) is used to denote the orientation or direction of a vector in the y-axis direction. It helps in expressing a vector in a three-dimensional space, where all vectors are referred relative to this basis, and can be written as the sum of the products of the component's magnitude and direction.

Step 3: Representation

If a vector \( \mathbf{A} \) has a y-component of say 'm', then this can be represented in the vector form as \( m\mathbf{j} \).

Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system is a foundational element in understanding mathematics and physics, particularly when it comes to graphing points, lines, and shapes in a two or three-dimensional space. Named after the French mathematician René Descartes, this system uses perpendicular axes that intersect at a central point known as the origin.

In a two-dimensional space, these axes are typically labeled as the x-axis (horizontal) and y-axis (vertical). However, to fully explore the realm of three-dimensional geometry and vectors, we introduce an additional z-axis, which represents depth and is perpendicular to both the x and y-axes. Together, these axes establish a three-dimensional grid where any point can be identified by its coordinates (x, y, z).

The convenience of this system lies in its ability to provide a clear method for locating and describing the position of points, lines, and planes. By utilizing the Cartesian coordinate system, one can easily decompose complex shapes and motions into their constituent parts along the x, y, and z directions.

Vector Components

To delve into the concept of vector components, let's first clarify what a vector is. A vector is a mathematical entity with both magnitude and direction, often represented by an arrow. In physics, vectors are crucial for describing quantities like velocity, force, and displacement.

When we decompose vectors into their respective components, we are essentially breaking them down into parts that align with the axes of our Cartesian coordinate system. For instance, in three-dimensional space, a vector can be split into x, y, and z components, each representing the vector's influence along the respective axis. These components are like building blocks, which when combined according to vector addition rules, reconstruct the original vector.

The process of finding vector components is often referred to as 'resolving a vector.' This involves determining the effect or magnitude of a vector along the axes directions. Without understanding vector components, analyzing physical problems such as projectile motion or forces acting on an object would become highly challenging.

Three-Dimensional Vectors

The transition from flat, two-dimensional analysis to the three-dimensional realm opens the door to a more comprehensive understanding of the space around us. In three-dimensional vectors, we extend our understanding beyond the flat plane, considering how objects move and forces act within a volumetric space.

Every three-dimensional vector can be uniquely represented by its components in the Cartesian coordinate system. These components are expressed in terms of unit vectors—essentially vectors with a magnitude of one—which define the direction along each axis. In the context of the exercise, the vector \( \mathbf{j} \) is one of these unit vectors, specifically aligned with the y-axis. The other two unit vectors commonly used are \( \mathbf{i} \) for the x-axis and \( \mathbf{k} \) for the z-axis.

By convention, if there's a vector \( \mathbf{A} \) with a y-component of magnitude 'm', we would write it as \( m\mathbf{j} \). This notation simplifies the expression and calculation of vectors in three-dimensional spaces. It allows us to visualize and calculate the effect of vectors in complex systems, such as understanding forces in mechanics or analyzing wind velocity in meteorology.