Scalars and Vectors in vector algebra
Scalars and vectors are important ideas in math and physics that help us talk about amounts and their features.
Scalar
A scalar is something that only needs magnitude to explain it, without saying which way it is going. Scalars are numbers that have a certain size, and they come with units of measurement. Scalars are things that we can measure that only have size or amount, like how much something weighs, how hot or cold it is, how long it takes to do something, how far away something is, how fast something is moving, how much energy something has, or how big something is. Scalar numbers can be added, subtracted, multiplied, or divided using basic math. In vector algebra, you can use one number (called a scalar) to make a new number (called a vector). You can also use a scalar with another scalar to make a new number.
Vector
A vector in vector algebra is a quantity that tells you both magnitude and direction. Vectors are shown by arrows. The size of the arrow tells how strong the vector is, and the direction shows which way it goes. Vectors are helpful for explaining things like how far something moves, how fast it goes, how strong a push is, how quickly it speeds up, how hard it is to stop something from moving, and how electrical energy flows. When you add vectors, you start by lining up their tails and then connecting the ends to get the total vector. To subtract a vector, you add the opposite of that vector. In Math, you can use different methods to multiply vectors. Cross and tensor products make new vectors while the dot product gives you just one number. A depiction of scalars and vectors is given in Figure 1.
Vectors in vector algebra can be shown by using their parts in different directions. In Cartesian coordinates, we can describe vectors using their x, y, and z parts. Unit vectors are special vectors like i, j, and k. They have a length of 1 and show you which way to go. They are commonly used to show directions in a graph. Knowing about vectors and the way they work is very important for solving problems in physics, engineering, computer science, and other types of work.
To put it simply, scalars and vectors are important when talking about amounts in different areas of science. Scalars only tell you how much of something there is, while vectors also tell you which way it is going. Scalars are numbers that can be added, subtracted, multiplied, or divided using mathematics. They can also be multiplied with other types of numbers called vectors or more scalars.
Different Vector Operations
In vector algebra, vectors can be combined, separated, stretched or shrunk, and combined with other vectors using math. It’s important to understand how to tell the difference between scalars and vectors and use them correctly to properly understand and study how things work in the physical world.
Vector Addition
Vector addition in vector algebra is when you put two or more vectors to make one resultant vector. Draw a line starting from the end of the first arrow to the top of the last arrow. The size of the final vector depends on how long the original vectors are, and the way it’s pointing is based on how those vectors are lined up.
Vector Subtraction
Vector subtraction means finding the answer when you take away one vector from another vector. You can do this by adding the opposite direction of the vector you’re taking away. To take away a vector, we turn it around and then add it to another vector.
Position Vector
A position vector tells us how far something has moved from one point to another. A vector is like an arrow that shows how far something has moved in space. It starts from a starting point and ends at a final point. The position vector is a way to measure how far apart two points are, and it shows which way they are in a straight line.
Distance Vector
Distance vector is a way to show how far apart and which way two points are in space. This thing is kind of like a position pointer, but it doesn’t start from one specific spot. It only shows how far away and which way the two points are from each other. The size of the distance line shows how far apart two points are, and the direction shows which way to go from one point to the other.
Both position and distance are important in different situations. They use them in physics, engineering, and navigation to talk about how far apart things are from each other. When we know how to add and subtract directions and measure distances in a particular direction, we can correctly show and study how objects move and relate to each other in the real world.
Vector Multiplication
There are two different ways to multiply vectors: dot product and cross product. These math actions work in different ways and give different answers.
Dot Product
The dot product of two vectors is a number that shows how much they are alike or pointing in the same direction. This means that to find something called the answer, you need to use two things called vectors and calculate their size and angle between them using multiplication and a thing called cosine. The dot product is about how one vector is shown on another one. This number helps you figure out if two lines are at right angles (it’s zero), going in the same direction (it’s a positive number), or going in opposite directions (it’s a negative number).
Cross Product
Multiplying two arrows gives a new arrow that is at a right angle to the flat space made by the first arrows. The size of a cross product is the result of multiplying the sizes of two vectors and multiplying it with the sine of the angle between them. To find the direction of the cross product, use your right hand. Curl your fingers from the first vector to the second. The direction your thumb points is the direction of the cross product. The cross-product works differently if you change the order of the vectors. This thing helps you measure how big a parallelogram is when made by two arrows pointing in different directions. It can also be used in physics to figure out how much something can turn and to find out what direction a flat surface is pointing in.
The dot product and the cross product are used in different fields for specific reasons. In physics, the dot product helps you figure out how much work is done, find the angle between vectors, and make projections. It is used in computer graphics to calculate lighting and shading. The cross product helps physicists calculate torque, magnetic field strength, and angular momentum. This is useful in engineering to solve problems with forces, speeds, and turning.
To put it simply, there are two types of vector multiplication in vector algebra: dot product and cross product. The dot product gives a number that shows how similar two vectors are, while the cross product creates a new vector that is at a right angle to the two original vectors. Two math functions have specific properties and are used in physics, engineering, and computer graphics to analyze and solve problems involving lines that meet at a point.
The slides of this Topic are given below and they can be taught in;
Electricity and Magnetism
Electromagnetic theory
Lecture 1: Vector Algebra 1
References
Cheng, D. K. (1989). Field and wave electromagnetics. Pearson Education India.
Ulaby, F. T., & Ravaioli, U. (2015). Fundamentals of applied electromagnetics (Vol. 7). Upper Saddle River, NJ: Pearson.
Ida, N. (2015). Engineering electromagnetics (Vol. 2). New York: Springer.
Further Readings
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