Vectors have many uses in real life and different areas of study. They are used to show direction and movement in physics, engineering, and navigation. Vector algebra and vector spaces help solve problems in math and science. Engineers use vectors to design machines and structures. The dot product of vectors is useful in finding angles and forces between objects. In this article, we will learn about the different ways vectors are applied in real-world situations, in engineering, and in mathematics.

Vector Introduction

A vector is a quantity that has both a size (called magnitude) and a direction. This means it tells us how much and which way something is moving or pointing. Common examples of vector quantities include displacement, force, and electric field intensity.

In math, vector algebra is a special area that follows logical rules to work with vectors. Geometrically, a vector looks like an arrow drawn from one point to another. For example, if you have a line going from point A to point B, and you add an arrow pointing from A to B, it becomes a vector and is written as →AB. This shows both the length of the line (magnitude) and the direction it goes. If the arrow points from B to A instead, it's written as →BA, and the direction is reversed. Vectors are used in physics and math to describe motion, forces, and more.

Vectors can be used by air-traffic controllers when tracking planes, by meteorologists when describing wind conditions, and by computer programmers when they are designing virtual worlds. In this section, we will present three applications of vectors that are commonly used in the study of physics: work, torque, and magnetic force.

Vector Calculus

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space represented by \(R^3\)

Vector calculus plays an important role in differential geometry and in the study of partial differential equations. Vector calculus also deals with two integrals known as the line integrals and the surface integrals.

Divergence and curl are two important operations on a vector field. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus. In addition, curl and divergence appear in mathematical descriptions of fluid mechanics, electromagnetism, and elasticity theory, which are important concepts in physics and engineering.

Application of Vector Calculus

It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.

Vector Calculus is used in

• Geodesics on a Surface
• Electric Field from Distributed Charge
• Plotting a Slice of a Vector Field
• To find the rate of change of the mass of a fluid flows.
• In rigid body dynamics in rectilinear and plane curvilinear motion along paths and in both rectangular.

Learn about Scalar Triple Product

Solved Example of Application of Vector Calculus in Engineering Mathematics

1. Find the angle between the tangents to the curve \(\vec{r}=3t\hat{i}+2t\hat{j}-t^3\hat{k}\) at the point t = ± 1.
2. Differentiating the given curve w. r. to t, we \({d\vec{r}\over{dt}}=3\hat{i}+2\hat{j}-3t^2\hat{k}\) which is the tangent vector to the curve at any point t.

Let \(\vec{x}\) & \(\vec{y}\) are the tangent vectors to the curve at t = 1 and t = -1 respectively, then

\(\vec{x}=2\hat{i}+2\hat{j}-3\hat{k} and \vec{y}=-2\hat{i}+2\hat{j}-3\hat{k}\)

Let 𝜃 be the angle between the tangents \(\vec{x}\) & \(\vec{y}\) then

\(cos\theta={\vec{x}.\vec{y}\over{|\vec{x}|.|\vec{y}|}}={(2\hat{i}+2\hat{j}-3\hat{k}).(-2\hat{i}+2\hat{j}-3\hat{k})\over{|2\hat{i}+2\hat{j}-3\hat{k}|.|-2\hat{i}+2\hat{j}-3\hat{k}|}}={-4+4+9\over{\sqrt{17}\sqrt{17}}}={9\over{17}}\)

Learn about Types of Vectors

Vector Space

In mathematics, physics, and engineering, a vector space is a set of objects called vectors, which may be added together and multiplied by numbers called scalars. Scalars are often real numbers, but some vector spaces have scalar multiplication by complex numbers or, generally, by a scalar from any mathematical field. The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector

Application of Vector Space

Application of vector space is required in Engineering and computer science. Vector spaces have many applications as they occur frequently in common circumstances, namely wherever functions with values in some field are involved.

• They are used in Fourier Transformation
• Vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors.
• Application of vector space in computer science: The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector space methods.
• Application of vector space in linear algebra: Quantum Mechanics is entirely based on it. Also important for time domain (state space) control theory and stresses in materials using tensors.
• In differential geometry, the tangent plane to a surface at a point is naturally a vector space whose origin is identified with the point of contact.

Learn about Coplanar Vectors

Vector Algebra

Vector algebra is specifically the basic algebraic operations of vector addition and scalar multiplication. Vector Algebra includes addition and subtraction of vectors, division and multiplication of vectors, along with dot product and cross product.

Application of Vector Algebra

The list below is some of the most common Applications of Vectors Algebra.

• In many physical situations, we often need to know the direction of a vector. For example, we may want to know the direction of a magnetic field vector at some point or the direction of motion of an object.
• Vector algebra is useful to find the component of the force in a particular direction.
• In kinematics to find resultant displacement vectors and resultant velocity vectors.
• In mechanics to find resultant force vectors and the resultants of many derived vector quantities.
• In electricity and magnetism to find resultant electric or magnetic vector fields.
• Application of vectors in physics: Vectors can be used to represent physical quantities. Most commonly in physics, vectors are used to represent displacement, velocity, and acceleration. Vectors are a combination of magnitude and direction and are drawn as arrows.

Learn more about Logarithmic Functions here.

Application of Resolution of Vectors in Daily Life

Application of Resolution of Vectors in Daily Life is as listed below:

Banking of Roads

A road at curves is elevated at the farther end of curvature. The angle of banking is Ф. The normal reaction from the ground is N. The vehicles are inclined to vertical by angle Ф. N cos Ф balances the weight mg of the vehicle along vertical lines. N sin Ф supplies the centripetal force along the radius of curvature. That determines the maximum speed of the vehicle to avoid slipping.

Learn more about Banking of Roads

Projectile Motion

A projectile (stone) thrown with an initial speed u at angle Ф with the horizontal, has a vertical component of (u sin Ф – g t) and the horizontal component of u cos Ф under components of vector.

Sharpening wooden pencil with a blade

We cut the pencil at an angle. The component of force in the direction perpendicular to the pencil cuts the pencil. The component of force in the direction parallel to the pencil removes the thin wooden part.

Earth’s magnetic field

Earth’s magnetic field has two components B and H: perpendicular to Earth’s surface and parallel to the surface.

Pendulum

The tension in the string has two components to balance the weight and to give the centripetal force.

Real Life Application of Parallelogram Law of Vectors

Let P and Q be two vectors acting simultaneously at a point and represented both in magnitude and direction by two adjacent sides OA and OD of a parallelogram OABD as shown in the figure.

Let θ be the angle between P and Q and R be the resultant vector. Then, according to the parallelogram law of vector addition, diagonal OB represents the resultant of P and Q.

The magnitude of resultant vector is given by the following formula

\(R = \sqrt{P^2+Q^2+2PQcos\theta}\)

\(\phi=tan^-({Qsin\theta\over{P+Qsin\theta}})\)

1. Two forces of magnitude 6N and 10N are inclined at an angle of 60° with each other. Calculate the magnitude of the resultant and the angle made by the resultant with 6N force.
2. Let P and Q be two forces with magnitude 6N and 10N respectively and θ be the angle between them. Let R be the resultant force.

So, P = 6N, Q = 10N and θ = 60°

We have,

\(R = \sqrt{P^2+Q^2+2PQcos\theta}\)

\(R = \sqrt{6^2+10^2+2\times6\times10cos60}\)

\(R = \sqrt{196}\)

R = 14

Common Examples Applications of Vector Quantities

The list below is some of the most common Applications of Vectors.

1. Navigation and GPS
   Vectors help determine exact position, direction, and shortest routes.
2. Physics and Engineering
   Used in analyzing forces, motion, and designing structures (bridges, buildings).
3. Aviation
   Pilots use vectors to calculate wind speed/direction and adjust flight paths.
4. Sports
   Players use direction and speed to control ball movement (e.g., football, cricket).
5. Robotics and Game Development
   Vectors control movement, direction, and collision detection in programming.
6. Electricity and Magnetism
   Used to represent and calculate electric and magnetic fields.
7. Mechanical Systems
   In machines, vectors determine torque and motion direction.

Real-Life Experiments Based on Vector

Here are some simple experiments that show how vectors are used in science and engineering:

1. Projectile Motion Experiment

This experiment helps us understand how objects move through the air. For example, if you throw a ball at different angles and speeds, you can measure how far it goes and how long it stays in the air. By using vectors, you can describe the ball's speed, direction, and how gravity affects it.

2. Force Table Experiment

In this experiment, students use a tool called a force table to study how different forces work together. They attach weights with strings at different angles and see how the forces balance. This helps them learn how to add forces using vector diagrams and find the direction of the total (resultant) force.

3. Magnetic Field Mapping Experiment

This experiment is used to study magnetic fields. A sensor is moved around a magnet to measure the strength and direction of the magnetic field at different points. These measurements are shown as arrows (vectors), helping us understand how magnetic forces work and how they spread out.

Work

In physics, the term work is used to describe the energy that is added to or removed from an object or system when a force is applied to it. From the experiment, it has been determined that work is maximized when the applied force is parallel to the motion of the object and that no work is done when the force is applied perpendicular to the motion. Therefore, the work done by a force can be described by the dot product of the force vector and the displacement vector.

Using Vector calculus we can find the formula for work. The formula for work: W = \(\vec{F}·\vec{d}\). This means that work is a scalar quantity. It is the dot product of two vectors.

Hence, W = \(Fdcos\theta\), Where \(\theta is the angle between force and displacement.

Solved Example: Application of dot product of vectors

1. Rahul turns a crank to lower a bucket of water into a well. Determine the total work done on the bucket if the weight of the bucket is 15 N and the tension force in the rope is 13 N. The bucket rises a distance of 4.5 m while he is cranking in a vertically downwards direction.
2. The rope does negative work on the bucket because the motion and the force are in opposite directions. If the force is measured in newtons and the displacement in meters, the work is measured in Joules.

= −58.5J

The weight force does positive work on the bucket because the motion and the force are in the same direction.

\(W_{weight}=\vec{F_{weight}}\vec{d}cosθ\)

= (15N) (4.5m) cos 0∘

= 67.5J

Total Work done =\(W_{rope} + W_{weight}\)= −58.5J + 67.5J = 9.0J

Magnetic Force

The force that a magnetic field exerts on a charged particle is strongest when the particle moves perpendicular to the field and the magnetic force on the particle is equal to zero when it moves parallel to the field. Therefore the magnetic force can be described using the cross-product of the field strength vector and the particle’s velocity vector: \(\vec{F}=q\vec{v}\times\vec{B}\) where \(\vec{F}\) is the force on the particle, q is the charge of the particle, \(\vec{v}\) is the velocity of the particle, and \(\vec{B}\) is the vector representing the magnetic field. If the velocity is measured in m/s and if the magnetic field is measured in tesla, the force will be measured in newtons, the metric base unit of force. This means that magnetic force is a vector quantity. It is the cross product of two vectors.

Hence, \(F = qvBsin\theta.\)

Solved Example: Application of cross product of vectors

1. Determine the magnetic force between two parallel conductors of length 1 m separated by 50 cm in air and carrying currents of 30 A in the same direction and opposite direction.

A: Given as,

\(i_1 = i_2\) = 30 A,

d = 0.5 m,

L = 1 m

The magnetic force between the two conductors is,

F = 0.36 × \(10^{-3}\) N

\( F_{21}={\mu_0I_1I_2L\over{2\pi d}}={4\pi\times 10^{-7}\times 30^2 \times 1\over{2\times\pi\times0.5}}={0.36\times10^{-3}}N\)

Application of Magnetic Vector Potential

The vector potential is defined to be consistent with Ampere’s Law and can be expressed in terms of either current i or current density j (the sources of magnetic field). In various texts this definition takes the forms

\(A={\mu_{0}i\over{4\pi}}\oint{\vec{dl}\over{r}}\)

Torque

When you lift a baseball off a table-top, you are exerting a force that moves the object as a whole. When you apply a force to a doorknob, you cause the door to rotate on its hinges. Scientists use the term torque to describe the force-like property that affects the rotation of an object. The torque can be described using the cross-product of the force vector and the lever arm, a vector pointing radially outward from the axis of rotation to the point where the force is applied to the object: \(\vec{tau}=\vec{r}\times\vec{F}\), where \(\vec{tau}\) is the torque, \(\vec{r}\) is the perpendicular distance, and \(\vec{F}\) is the applied force. This means that torque is a vector quantity. It is the cross product of two vectors.

Hence, \(\tau = Frsin\theta.\)

Solved Example: Application of cross product of vectors

1. If the force applied is perpendicular to the handle of the spanner as shown in the diagram, find the (i) torque exerted by the force about the centre of the nut, (ii) direction of torque and (iii) type of rotation caused by the torque about the nut.
2. Arm length of the spanner, r = 15 cm = \(15×10^{−2}m\)

Force, F = 2.5 N

Angle between r and F, θ = 90

\(\tau = Frsin\theta\)

\(\tau=2.5\times15\times10^{−2}\times{sin90}\)

\(\tau=37.5\times10^{−2}Nm\)

As per the right-hand rule, the direction of the torque is out of the page. The type of rotation caused by the torque is anticlockwise

Daily Life Applications of Vectors

Following are the everyday applications of vectors in daily life

• Navigating by air and by boat is generally done using vectors.
• Planes are given a vector to travel, and they use their speed to determine how far they need to go before turning or landing. Flight plans are made using a series of vectors.
• Sports instructions are based on using vectors. For example, wide receivers playing American football might run a route where they run seven meters down the field before turning left 45 degrees and running in that direction. Sports commentary also depends on vectors. Only a few sports have fields with grids, so discussions revolve around the direction and speed of the player.

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