Find \((a)\) the vector component of \(\mathbf{F}=10 \mathbf{a}_{x}-6 \mathbf{a}_{y}+5 \mathbf{a}_{z}\) that is parallel to \(\mathbf{G}=0.1 \mathbf{a}_{x}+0.2 \mathbf{a}_{v}+0.3 \mathbf{a}_{z} ;(b)\) the vector component of \(\mathbf{F}\) that is perpendicular to \(\mathbf{G} ;(c)\) the vector component of \(\mathbf{G}\) that is perpendicular to \(\mathbf{F}\).

Electromagnetics is a fundamental branch of engineering that involves the study of electromagnetic fields, waves, and their interactions with matter. It is deeply rooted in Maxwell's equations, which describe how electric and magnetic fields are generated and altered by each other, as well as by charges and currents. Understanding vector components is essential in this field, as electromagnetic quantities such as electric and magnetic field strengths are often represented as vectors.

For students diving into electromagnetics, mastering the analysis of vector components allows them to predict behaviors such as the propagation of electromagnetic waves, the force on charged particles, and the distribution of fields around conductors. Exercises like the one provided, where vector components are to be found that are parallel and perpendicular to other vectors, solidify foundational skills necessary in advanced applications like antenna design, wave propagation, and signal analysis.

In practicing with vectors and their components, students not only learn to manipulate mathematical representations but also develop an intuition about the physical concepts that the mathematics represents. This aligns with the goal of electromagnetics education to bridge the gap between theoretical principles and practical engineering challenges.

Vector analysis is a cornerstone of engineering, especially when dealing with physical phenomena. In the context of engineering problems, vectors describe quantities that have both magnitude and direction, such as forces, velocities, and accelerations. The problem provided exemplifies how engineers often need to break down a vector into its components—that is, the parts of the vector that align with a chosen set of axes or with other vectors.

Engineering vector analysis entails using various mathematical tools to understand and manipulate these vector components. Techniques such as dot products, cross products, and vector projections become indispensable. For instance, the dot product helps in finding the angle between vectors or in calculating the component of one vector in the direction of another, known as the projection. These calculations are not just academic; they are crucial for tasks ranging from the analysis of structural loads to the design of electrical circuits and control systems.

Therefore, step by step solutions to problems, such as vector decomposition, serve as vital practice for students to gain proficiency in these analytical tools that they will use as professionals in engineering fields. The importance lies in understanding both the mathematical procedure and the physical significance of each vector component's role.

Understanding Vector Projection

The vector projection is a fundamental operation in vector analysis, where one vector is projected onto another. Essentially, it answers the question: If you were to cast a shadow of one vector onto another, what would that shadow look like?

As demonstrated in the provided exercise, to calculate the vector projection of \(\mathbf{F}\) onto \(\mathbf{G}\), you need the dot product of vector \(\mathbf{F}\) and the unit vector in the direction of \(\mathbf{G}\), denoted as \(\mathbf{\text{hat}{G}}\). The result tells us the length of the 'shadow', which we then scale according to the direction of \(\mathbf{G}\). This process yields the component of \(\mathbf{F}\) that lies along the same line as \(\mathbf{G}\).

Vector projection has practical importance in various fields, such as calculating work done by a force acting at an angle to the displacement or in computer graphics, where it is used to render 3D objects onto a 2D screen. Understanding vector projections is therefore not just a mathematical exercise; it is a tool that has real-world applications in engineering and physics.

Finding the Perpendicular Vector Component

An equally crucial concept is the vector component that is perpendicular to another vector. While the projection gives us the component that lies along the direction of another vector, the perpendicular component is orthogonal, meaning it forms a right angle with the other vector.

As in our problem, the perpendicular component of \(\mathbf{F}\) with respect to \(\mathbf{G}\) is found by subtracting the projection from the original vector \(\mathbf{F}\). This resulting vector points in a direction that does not share any 'overlap' with \(\mathbf{G}\), and its magnitude represents the shortest distance from the tip of \(\mathbf{F}\) to the line along \(\mathbf{G}\).

Perpendicular components have significant implications in areas such as force decomposition, where understanding the normal force is key for analyzing static equilibrium, or in telecommunications, where perpendicular vectors can represent signals that are independent of each other. Ensuring that students can compute and understand such components is central to training competent engineers and physicists who can solve complex spatial problems.