Module 1 Vector and Scalar Quantity PDF

Summary

This document explains the difference between scalar and vector quantities. It gives examples of each type. It also describes methods for determining the resultant vector.

Full Transcript

**Module 1** **Vector and Scalar Quantity** **Week Covered: Week 1 and Week 2** **Lesson Objectives:** - Have a deeper understanding of vector and scalar quantities - Characterize difference between scalar and vector quantities - Identify and differentiate the different methods to determ...

**Module 1** **Vector and Scalar Quantity** **Week Covered: Week 1 and Week 2** **Lesson Objectives:** - Have a deeper understanding of vector and scalar quantities - Characterize difference between scalar and vector quantities - Identify and differentiate the different methods to determine the resultant vectors of given quantities using mathematical equations and graphical method. **Vector and Scalar Quantities** In mathematics and physics, we have physical quantities which can be categorized in two ways, namely - Scalar Quantity - Vector Quantity In this article, let us discuss what are vector and scalar quantities with examples. **Scalar Quantity Definition** The physical quantities which have only magnitude are known as scalar quantities. It is fully described by a magnitude or a numerical value. Scalar quantity does not have directions. In other terms, a scalar is a measure of quantity. For example, if I say that the height of a tower is 15 meters, then the height of the tower is a scalar quantity as it needs only the magnitude of height to define itself. Let's take another example, suppose the time taken to complete a piece of work is 3 hours, then in this case also to describe time just need the magnitude i.e. 3 hours. **Scalar Quantity Examples** Other examples of scalar quantities are mass, speed, distance, time, energy, density, volume, temperature, distance, work and so on. **Vector Quantity Definition** The physical quantities for which both magnitude and direction are defined distinctly are known as vector quantities. For example, a boy is riding a bike with a velocity of 30 km/hr in a north-east direction. Then, as we see for defining the velocity, we need two things, i.e. the magnitude of the velocity and its direction. Therefore, it represents a vector quantity. **Vector Quantity Examples** Other examples of vector quantities are displacement, acceleration, force, momentum, weight, the velocity of light, a gravitational field, current, and so on**.** **Vector Representation** Let us have a look at the [line segment](https://byjus.com/maths/line-segment/) drawn below. A vector quantity always has a starting point and an endpoint. The two endpoints of the given line segment are distinguishable as and. It represents a directed line segment ![](media/image2.png) **Characteristics of Vectors** The characteristics of the vectors are as follows: - Vectors possess magnitude as well as the direction - It does not obey the ordinary law of algebra - Either the magnitude or direction change or both change **Scalar and Vector Quantity Example** Question:  Find out the scalar and vector quantity from the given list. Force, Speed,  Electric field, Angular Momentum, Magnetic Moment, Temperature, Linear Momentum, Average Velocity. Solution: From the given list, - Scalar Quantities -- Speed, Temperature. - Vector Quantities -- Force, Electric field, Angular Momentum, Magnetic Moment, Linear Momentum, Average Velocity. Now we are familiar with what are vectors and scalars. Now if somebody asks if acceleration is a vector or a scalar, we can easily tell that it's a vector because it has direction as well as magnitude. Similarly, when asked if the distance is a vector or scalar, it is quite evident that as distance has only magnitude, it is a scalar quantity. **Methods in Determining the Resultant Vector** In physics, a quantity, such as mass, length, or speed that is completely specified by its magnitude and has no direction is called a scalar*.* A vector, on the other hand, is a quantity possessing both magnitude and direction. A vector quantity is typically represented by an arrow-tipped line segment. The length of the line, drawn to scale, represents the magnitude of the quantity. The direction of the arrow indicates the direction of the vector. Not only can vectors be represented graphically, but they can also be added graphically. For one dimensional vector addition, the first vector is placed on a number line with the tail of the vector on the origin. The second vector is placed with its tail exactly on the arrow head of the first vector. The sum of the two vectors is the vector that begins at the origin and ends at the arrow head of the final added vector. ![](media/image4.png) **Adding Vectors in Two Dimensions** **Vector Addition and Subtraction- Analytical Methods** Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known. **Resolving a Vector into Perpendicular Components** ![](media/image6.png)Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like AA in Figure 3.3.1, we may wish to find which two perpendicular vectors, Ax and Ay, add to produce it. Figure 3.3.1: The vector AA, with its tail at the origin of an x, y-coordinate system, is shown together with its x- and y-components, Ax and Ay. These vectors form a right triangle. The analytical relationships among these vectors are summarized below. ![](media/image8.png) A graph of a function Description automatically generated with medium confidence ![](media/image10.png)**Calculating a Resultant Vector** **Adding Vectors Using Analytical Methods** ![](media/image12.png) **Example:** Add the vector A to the vector BB shown in Figure, using perpendicular components along the *x*- and *y-*axes. The *x*- and *y*-axes are along the east--west and north--south directions, respectively. Vector A represents the first leg of a walk in which a person walks 53.0m in a direction 20.0º north of east. Vector B represents the second leg, a displacement of 34.0m in a direction 63.0º north of east. **Solve the following:** ![A black text on a white background Description automatically generated](media/image14.png) **Component Method** 1. Find the components of each vector a. Force of 300 N at 30^0^ above x axis b. Force of 200 N at 210 below the x axis 2. Calculate the magnitude and direction of the resultant vector that is formed when taking the sum of the two vectors as shown: F1= 200 at 30^0^ and F2 = 300 at 45^0^ 3. Five people are pulling the box 1^st^ = 6N East 2^nd^ = 7 N 40^0^ North of East 3^rd^ = 9 N South 4^th^ = 4 N 30^0^ North of West 5^th^ = 8N 70^0^ South of West 4. A map of South Korea suggests that Daegu is 155 km in a direction 22.5 North of East from Gwangju. The same map show that Seoul is 230 km in a direction 36^0^ West of North from Daegu. Modeling the earth as flat, use the information to find the displacement from Gwangju to Seoul. 5. Find the resultant vector A 15N at 60 degrees B 10 N at 315 degrees C 12 N at 160 degrees D 8 N at 270 degrees E 9N at 180 degrees F 2N at 296 degrees

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