Vector Analysis in FE Electrical Exam

Vector Analysis is a crucial electrical engineering concept involving using vectors to describe and solve complex problems. In the FE Electrical Exam, mastering Vector Analysis is essential for tackling challenging issues involving magnetic fields, electric circuits, and other electromagnetics-related topics.

Beyond the exam hall, Vector Analysis is a fundamental skill that plays a vital role in many areas of electrical engineering. A strong understanding of Vector Analysis is essential for success, from designing power systems to developing advanced technologies.

This blog will dive into Vector Analysis and explore its importance in the FE Electrical Exam and beyond. We’ll cover the basic concepts, including vector operations, coordinate systems, and vector fields, and provide tips and tricks for mastering this essential skill. 

Whether you’re a student preparing for the FE Electrical Exam or an electrical engineer looking to improve your skills, this study guide will provide valuable insights into Vector Analysis.

Importance of Vector Analysis in Electrical Engineering

Below are five points that highlight the importance of Vector Analysis in Electrical Engineering:

  • Unwrapping Electromagnetic Fields – Vector Analysis is essential for describing and analyzing electromagnetic fields. For example, engineers design an induction motor using Vector Analysis to understand the interactions between magnetic fields and electric currents. Engineers can better understand their behavior and design more efficient systems by manipulating vectors to represent these fields.
  • Solving Complex Circuits – Vector Analysis is crucial for solving complex circuit problems. For example, engineers may use Vector Analysis to determine the voltage and current in each circuit branch and identify any potential issues, such as overloaded components. This enables them to design more efficient and reliable circuits.
  • Crafting Modern Communication Systems – Engineers use Vector Analysis to understand how signals are transmitted and received in developing communication systems. For example, Vector Analysis can be used to analyze the phase and amplitude of radio waves, allowing engineers to design antennas that can receive and transmit signals more efficiently.
  • Modeling Mechanical Systems – Vector Analysis is not limited to electrical systems. It is also essential for modeling mechanical systems, such as machines and structures. For example, engineers may use Vector Analysis to calculate the forces acting on a machine and design it to withstand these forces.
  • Power Systems Design and Analysis – Vector Analysis is crucial for designing power systems, including power grids and distribution networks. Engineers use Vector Analysis to determine the optimal placement of transformers, generators, and other components to ensure power is distributed efficiently and reliably.

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Vector Basics

A vector is a mathematical object that has both magnitude and direction. In physics and engineering, vectors describe physical quantities such as velocity, acceleration, force, and electric field.

A vector can be represented by an arrow, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector.

Electric field, magnetic field, force, torque, displacement, and all other physical quantities having direction are considered vectors because changing the direction can influence the result of adding or multiplying these physical quantities.

Let’s explain it in the context of distance (scaler quantity) and displacement (vector quantity).

Consider a person traveling 3 km to the east and 4 km to the north. We can represent these distances as vectors OA and AB, respectively, where O is the starting point and B is the final position reached by the person.

Vector OC can represent the person’s displacement, the shortest distance in a straight line from the starting point O to the final position B.

We simply add both distances to find the distance traveled, as the distance is a scaler quantity and has nothing to do with directions.

Distance = OA + AB = 3+4 = 7 Kms

But is the person 7 Km far from the starting point? Not. As he moved to the east, then north. His motion represents the rectangular components of a triangle. So his displacement (his shortest distance from the displacement) will be found by Pythagoras’ theorem as:

displacement = |OC| = √(3² + 4²) = √(9 + 16) = √25 = 5 km

Vector rectangular components

We use Pythagoras’ theorem because the distances are vectors in rectangular forms.

So, the person’s displacement expressed as the resultant vector length is 5 km, the shortest distance between two points in a specific direction. This is where direction comes into the frame and changes the course of your mathematical operations, making vectors different from scaler quantities having no direction.

Types of Vectors

The table below represents the types of vectors you must know before mastering the Vector Analysis concept in the FE Electrical Exam.

Type of VectorDefinition
Position vectorA vector that describes the position of a point in space relative to a fixed reference point or origin.
Unit vectorA vector with a magnitude of 1 is used to indicate direction.
Negative vectorA vector with the same magnitude as another vector but points in the opposite direction.
Zero vectorA vector with a magnitude of 0 is represented by $\vec{0}$.
Normalized vectorA vector that has been scaled to have a magnitude of 1 while maintaining its direction.
Parallel vectorsVectors that have the same direction or are scalar multiples of each other.
Orthogonal vectorsVectors that are perpendicular to each other and have a dot product of 0.
Projection vectorThe projection of one vector onto another is the component of the first vector that lies along the direction of the second vector.

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Vector Representation

In vector notation, consider a vector A represented as follows:

A = 2i + 3j + 4k

Here, i, j, and k are unit vectors in the x, y, and z directions. Each of these unit vectors has a magnitude of 1 and represents a specific direction in three-dimensional space.

So, in the vector A = 2i + 3j + 4k, the coefficient 2 in front of the i unit vector means that vector A has a component of 2 units in the x-direction (vector length for x-component). The coefficient 3 in front of the j unit vector indicates that vector A has a vector length of 3 units for the y-component (in the y-direction). 

Similarly, the coefficient 4 in front of the k unit vector means that vector A has a vector length for the z-component of 4 units (in the z-direction).

Therefore, the vector A is a three-dimensional vector with components in the x, y, and z directions. The unit vectors i, j, and k represent the coordinate axes in three-dimensional space and describe the direction and magnitude of three-dimensional vectors.

Addition and subtraction of vectors

To perform the addition of vectors A and B, we simply add their corresponding components:

A + B = (2i + 3j + 4k) + (4i – 6j + 2k)

= (2 + 4)i + (3 – 6)j + (4 + 2)k

= 6i – 3j + 6k

So, the vector sum of A and B is 6i – 3j + 6k.

To perform the subtraction of vector B from A, we simply subtract their corresponding components:

A – B = (2i + 3j + 4k) – (4i – 6j + 2k)

= (2 – 4)i + (3 + 6)j + (4 – 2)k

= -2i + 9j + 2k

So, the vector difference of A and B is -2i + 9j + 2k.

Add/Subtract vectors are fundamentally the same operations. Subtracting a vector B from A means adding a negative vector of vector B to vector A. A negative vector of B is a vector that has the same magnitude as B but points in the opposite direction. In other words, the negative vector of B is -B.

When we subtract vector B from A, we are essentially adding the negative vector of B to A:

A – B = A + (-B)

This is because adding a negative vector is the same as subtracting the corresponding positive vector. In other words, the negative of a vector reverses its direction without changing its magnitude.

So, we can write the vector difference of A and B as:

A – B = (2i + 3j + 4k) + (-4i + 6j – 2k)

= 2i – 3j + 2k

Therefore, subtracting vector B from A is the same as adding the negative of B to A.

Dot and cross product of vectors (Multiply/Divide Vectors)

The dot product, or the scalar product, is a binary operation that takes two vectors and returns a scalar. The dot product of two vectors is the product of their magnitudes and the cosine of the angle between them. It is denoted by a dot (.) between the two vectors. The dot product is commutative, meaning that the order of the operands does not matter.

Mathematically

The dot product of two vectors A and B is given by:

A . B = |A| |B| cosθ

The dot product is helpful in many applications, such as calculating the work done by a force in moving an object and determining the angle between two vectors. It is also used to find the projection of one vector onto another vector.

The cross product, also known as the vector product, is a binary operation that takes two vectors and returns a third vector perpendicular to both input vectors. A cross denotes the cross product between the two vectors.

The magnitude of the cross product is equal to the product of the magnitudes of the two input vectors multiplied by the sine of the angle between them.

right-hand rule

The direction of the resultant cross product can be found by using the right-hand rule, which states that if the fingers of the right hand are curled in the direction of the first vector and then rotated towards the second vector (angle from vector A to B), the thumb points in the direction of the cross product (Perpendicular to the plane of A & B).

Mathematically

The cross product of two vectors, A and B, is given by:

A x B = |A| |B| sinθ n

Where n shows the unit vector in the direction of the resultant vector of A x B.

The cross-product is helpful in many applications, such as calculating the torque produced by force on a rotating object and determining the area of a parallelogram spanned by two vectors.

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Vector Algebra

Dealing vector functions and equations as algebraic equations with the same algebraic rules and principles for vector addition/subtraction, vector multiplication/division, etc., falls under the umbrella of Vector Algebra.

Let’s discuss this in further detail.

Vector algebraic operations

Vector algebraic operations include addition, subtraction, and scalar multiplication. Vector addition involves adding the corresponding components of two vectors to create a new vector. For example, if we have two vectors A = (1, 2, 3) and B = (4, 5, 6), then the sum of A and B is A + B = (1 + 4, 2 + 5, 3 + 6) = (5, 7, 9). 

Vector subtraction is similar to vector addition but instead involves subtracting the corresponding components of two vectors. Scalar multiplication involves multiplying a vector by a scalar value, which multiplies each vector component by the scalar value.

For example, if we have a vector A = (1, 2, 3) and a scalar value c = 2, then the scalar multiple of A and c is cA = (2, 4, 6).

Vector differentiation

Vector differentiation involves finding the derivative of a vector function. This is similar to finding the derivative of a scalar function but involves taking the derivative of each component of the vector function.

For example, if we have a vector function f(t) = (t^2, 2t, t + 1), then the derivative of f(t) is f'(t) = (2t, 2, 1). The derivative of a vector function can be used to find the tangent vector of a curve in space.

Vector integration

Vector integration involves finding the integral of a vector function. This is similar to finding the integral of a scalar function but involves integrating each component of the vector function.

For example, if we have a vector function f(t) = (t^2, 2t, t + 1), then the integral of f(t) concerning t is F(t) = (1/3)t^3 + t^2 + t + C, where C is the constant of integration. Vector integration can be used to find the displacement vector of an object moving in space.

Line integrals

Line integrals involve integrating a vector function along a curve in space. This can be used to find the work done by a force along a curved path.

For example, if we have a force field F(x, y, z) = (2x, 3y, 4z) and a curve C defined by x = t^2, y = t, z = 2t, where 0 ≤ t ≤ 1, then the line integral of F along C is ∫ F(x, y, z) · dr, where dr is the differential of the curve, which is given by dr = (2t, 1, 2) dt. The line integral can be evaluated as ∫ F(x, y, z) · dr = ∫ (4t^3, 3t^2, 8t) · (2t, 1, 2) dt = 12.

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Vector Calculus

Vector Calculus

Vector Calculus is used to apply calculus operations on vector functions to calculate the gradient, area, and tangent of geometric figures involving planes, curves, etc. Let’s discover this in more detail.

A gradient of a scalar field

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the maximum rate of increase of the scalar field, and its magnitude is the rate of increase in that direction.

Mathematically, the gradient of a scalar field f(x, y, z) is denoted by ∇f(x, y, z), where ∇ is the gradient operator, and is defined as follows:

∇f(x, y, z) = (∂f/∂x) i + (∂f/∂y) j + (∂f/∂z) k

For example, consider the scalar field f(x, y, z) = x^2 + y^2 + z^2. The gradient of f is:

∇f(x, y, z) = 2x i + 2y j + 2z k

It’s the same as finding the gradient of a vector by differentiating it as we read above but applying it to the scaler field followed by assigning unit vectors with respective variables.

The partial derivative used here shows that we are differentiating each variable w.r.t to the same variable to avoid null result for the other variables. For example, differentiating the entire equation w.r.t x will result in the answer as

d/dx (x,y,z) = 2x i + 0 + 0 = 2x I

Divergence of a vector field

The divergence of a vector field measures the extent to which the vector field flows out or in at a given point. Mathematically, the divergence of a vector field F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) is denoted by div(F) or ∇ · F, and is defined as follows:

div(F) = ∇ · F = (∂P/∂x) + (∂Q/∂y) + (∂R/∂z)

For example, consider the vector field F(x, y, z) = (x^2, y^2, z^2). The divergence of F is:

div(F) = ∇ · F = (2x) + (2y) + (2z) = 2(x + y + z)

Curl of a vector field

The curl of a vector field measures the tendency of the vector field to rotate about a point. Mathematically, the curl of a vector field F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) is denoted by curl(F) or ∇ x F, and is defined as follows:

curl(F) = ∇ x F = (Ry – Qz) i + (Pz – Rx) j + (Qx – Py) k

For example, consider the vector field F(x, y, z) = (yz, xz, xy). The curl of F is:

curl(F) = ∇ x F = (0 – z) i + (0 – y) j + (x – x) k = -zi – yj

Stoke’s Theorem

Stoke’s theorem is a fundamental theorem of vector calculus. It relates the curl of a vector field to the line integral of the vector field around a closed curve in a plane. Stoke’s theorem is expressed mathematically as follows:

∫_C (curl F)·dS = ∫_S F·dr

C is a closed curve in a plane that bounds the surface S, and F is a vector field. The left-hand side of the equation is the line integral of the curl of F around the closed curve C, and the right-hand side is the surface integral of F over the surface S bounded by the curve C.

For example, consider a vector field F = (x^2 + y^2) i + (x – y) j + z k. Let C be the circle of radius 2 centered at the origin in the xy-plane, oriented counterclockwise, and S be the portion of the plane z = 1 that lies inside the circle. Then, the curl of F is:

curl F = (∂F_z/∂y – ∂F_y/∂z) i + (∂F_x/∂z – ∂F_z/∂x) j + (∂F_y/∂x – ∂F_x/∂y) k

= 2 j Therefore, by Stoke’s theorem, the line integral of F around the circle C is equal to the surface integral of the normal component of F over the surface S:

∫_C F·dr = ∫_S (F·n) dS

= ∫_S (0 – j) dS

= -∫_S j dS

= -π(2^2) = -4π

where n is the unit normal vector to S.

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Divergence Theorem

The divergence theorem relates the flux of a vector field through a closed surface to the volume integral of the divergence of the vector field inside the surface. The divergence theorem is expressed mathematically as follows:

∫_S F·dS = ∫_V (div F) dV

where S is a closed surface that bounds the volume V, F is a vector field, and div F is the divergence of F.

For example, consider the vector field F = x^2 i + y^2 j + z^2 k, and the closed surface S that bounds the unit cube centered at the origin. Then, the divergence of F is:

div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

= 2x + 2y + 2z

Therefore, by the divergence theorem, the flux of F through the surface S is equal to the volume integral of the divergence of F inside the cube:

∫_S F·dS = ∫_V (div F) dV

= ∫_0^1 ∫_0^1 ∫_0^1 (2x + 2y + 2z) dx dy dz

= 3

This means that the net flow of the vector field out of the unit cube is 3.

Applications of Vector Analysis in Electrical Engineering

The concepts you studied above for Vector Analysis in the FE Electrical exam are known for a broad range of applications in the Electrical engineering field.

Below are some of the critical applications of vector analysis in electrical engineering.

  • Maxwell’s equations – Vector analysis is used extensively in studying Maxwell’s equations, which describe the behavior of electromagnetic waves. These equations are crucial for designing and operating various electrical and electronic devices, such as antennas, transmitters, and receivers.
  • Electrostatics – Vector analysis plays a vital role in the study of electrostatics, which deals with the behavior of static electric charges. In electrical engineering, electrostatics is vital for designing and analyzing capacitors, which store electric charge, and for modeling electric fields around conductors.
  • Electromagnetic waves – Vector analysis is essential in studying electromagnetic waves. These waves comprise a composite field of perpendicular electric and magnetic fields propagating through space. Electrical engineers use vector analysis to design and analyze electromagnetic wave-based systems, such as wireless communication and radar systems.
  • Electromagnetic induction – Vector analysis is used to understand the phenomenon of electromagnetic induction, which is the generation of an electric current in a conductor due to a changing magnetic field. This phenomenon is the basis of various electrical and electronic devices, such as transformers, generators, and motors.
  • Vector calculus techniques – Electrical engineers use vector calculus techniques, such as gradient, divergence, and curl, to analyze and design electrical systems and devices. For example, the gradient is used to calculate the rate of change of a scalar field, such as the electric potential. D divergence and curl are used to calculate the behavior of electric and magnetic fields around conductors and devices.
  • Circuit analysis – Vector analysis determines complex circuits’ voltage and current relationships. For example, Kirchhoff’s laws of circuit analysis are based on vector analysis principles.
  • Signal processing – Vector analysis techniques are also used to analyze and manipulate various types of signals, such as audio and video signals. For example, Fourier analysis, which is based on vector analysis principles, is used to analyze and manipulate signals in the frequency domain.

Benefits and Impact: Vector Analysis as a Powerful Tool in Electrical Engineering

Vector analysis transcends simple calculations in electrical engineering. It is a powerful mathematical framework that unlocks efficient problem-solving, design optimization, and a unified approach to understanding diverse electrical phenomena. Some impacts of vector analysis include:

Problem-Solving Efficiency: Electrical engineering involves complex systems involving forces, fields, and energy transfer. Vector analysis allows engineers to model these phenomena mathematically with tools like gradient, divergence, and curl. For instance, calculating the forces acting on a motor’s rotor or analyzing the propagation of electromagnetic waves in an antenna becomes tractable through vector analysis. This mathematical framework simplifies complex problems and paves the way for efficient solutions.

Design Optimization: Optimizing electrical components and systems is crucial for performance and efficiency. Vector analysis empowers engineers to achieve this goal. For instance, vector analysis helps determine the most efficient configuration for magnetic field interaction with the rotor when designing a motor. Similarly, vector analysis aids in designing antennas and creates a compact and practical design for transmitting or receiving electromagnetic waves. By enabling the mathematical modeling of various design parameters, vector analysis becomes an invaluable tool for optimization.

Unified Approach: The beauty of vector analysis lies in its ability to offer a unified language for describing a vast array of electrical phenomena. Electric and magnetic fields, forces acting on charged particles, and even power flow in circuits – all can be expressed and manipulated using the principles of vector calculus. This unified approach allows engineers to transition seamlessly between different areas within electrical engineering. For example, understanding a concept like the curl becomes applicable to magnetic fields and the flow of current in a conductor. This fosters a deeper understanding of the interconnected nature of various electrical principles.

In conclusion, vector analysis is far more than just a set of mathematical tools. It empowers electrical engineers to tackle complex problems efficiently, optimize designs for performance, and seamlessly navigate the interconnected world of electrical phenomena.

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Conclusion

Vector analysis is a crucial tool for electrical engineers, and its importance cannot be overstated. The concepts of vectors, vector algebraic operations, and vector calculus are fundamental to studying and designing electrical and electronic systems.

From understanding the behavior of electromagnetic waves to analyzing complex circuits, vector analysis in the FE electrical exam is crucial to master from all aspects discussed above.

While preparing for the FE electrical exam, you must fully grasp vector analysis and its applications. At “Study for FE,” you’ll find a comprehensive platform that offers everything you need to succeed on the exam, including study materials, practice questions, and expert guidance.

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wasim-smal

Licensed Professional Engineer in Texas (PE), Florida (PE) and Ontario (P. Eng) with consulting experience in design, commissioning and plant engineering for clients in Energy, Mining and Infrastructure.