Mathematics and physics consist of various quantities. We come across a lot of different entities that have different kinds of properties. Most of the entities that we deal with in mathematics and science only consist of magnitude. Few quantities that consist of the only magnitude are speed and time. All those quantities that are only consists of magnitude are known as the scalar quantities. Scalar quantities follow normal operations of algebra. But apart from these quantities, there is one more type of quantity, that not only include the magnitude but also includes direction. Quantities including both direction and magnitude are commonly known as vector quantities. These quantities do not follow basic operations of algebra and have special operations that need to be followed to solve two vectors. Two such operations of vectors are vector product and dot product. In this article, we will discuss in detail vectors and dot products.
Vectors: These are the quantities that not only consist of magnitude but also direction. Vectors always consist of two points. One point is known as the initial point and refers to the starting of the vector whereas the other point is known as a terminal point, it is the point where the vector ends or it is the final position of the vector. Vectors are generally represented by an arrow sign that has a tail and head. Vectors follow a lot of operations such as the addition and subtraction of two vectors.
One of the major operations that are performed on two or more vectors is multiplication. Vector multiplication is not like normal multiplication that we perform in algebra. Vector multiplication is further divided into two methods. The first method is a cross product when two or more vectors are multiplied in this way, then their product is also a vector quantity but that is not the case with the other method of multiplication. The second method is the dot product. Let us discuss the dot product in detail.
Dot product: Dot product is one of the methods of multiplying two vectors. When we multiply two vectors the resultant is always a scalar quantity. As the result of the dot product is a scalar quantity, this method is also called the scalar product. One important thing to note about a scalar product is that the resultant of the dot product of two vectors always lie in the same plane, in which the two vectors lie. As the resultant we are getting only consists of magnitude, it can be a positive or negative real number.
Let us consider two vectors that are denoted by a and b. When their dot product is done it is generally represented as a.b, the dot between the two represents dot product. Now, this dot product of a and b is equal to the product of the magnitude of a, the magnitude of b and cos(theta), where theta is the angle made by the two vectors. It should be noted that we can find the angle between the two vectors if we know the magnitude of the two vectors and also their dot product. One just needs to modify the above formula to get cos(theta). Now one can easily get theta from it and that will be the value of the angle between the two vectors.
In the above article discussion on vectors and dot products have been done in the best possible way. This discussion focuses on all the important topics related to vectors. It is a topic of great importance, students must grab these topics clearly, without having any doubt in their minds. These topics form the foundation for higher studies, if one is clear in these topics then he or she will not face any problem in the higher studies. Vectors importance increases rapidly in physics as there are quantities like velocity and displacement that are of great importance and all of them are vectors. If students face any problem in understanding such math-related topics, then they should consider studying from Cuemath. Cuemath explains such math concepts to students in an easy way. They are known all over the world for their unique way of teaching.