Vectors are one of the most fundamental tools in mathematics and are of great importance in many different fields. They contain a wealth of useful information concerning the direction and magnitude of a certain quantity. The applications of vectors are critical in both science and engineering, and even more important in pivotal fields such as electromagnetic theory, fluid dynamics, and AC circuit analysis, just to name a few. Our conventional sense of what an “angle” and “length” are needs a bit of revision. This is because we have always primarily used the concepts of “length” and “angle” in a two-dimensional perspective. However, the true purpose of vector angles extends to three dimensions and beyond. For us to understand the angle between the two vectors, we must have some prior knowledge about trigonometry and the basic operations of vectors.
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Well, trigonometry is simple in that it deals with the study of triangles and their attributive properties, such as length and angles. Of all the triangles, the right-angle triangle is the most special of them all. The right-angle triangle consists of three parts that are called the adjacent, opposite and hypotenuse. The adjacent, which can be seen in the image below, is the side next to the angle theta. The opposite is the side opposite to the angle theta. Finally, the longest side of the triangle is the hypotenuse.
There are three main functions in trigonometry: the sine, cosine, and tangent. These are nothing but the ratios of one side of a triangle to another side of the triangle. Irrespective of the size of the triangle, whether it is big or small, as long as the ratio of two sides remains constant, the angle remains the same.
An Understanding of Vectors
A vector is considered to be an object that has both magnitude and direction. A geometric representation of a vector can be a directed line segment whose length is the magnitude of the vector. The direction of a vector usually runs from the tail to the head.
The three primary operations that one must understand when dealing with any form of vector mathematics is vector addition, vector subtraction and vector multiplication. The addition of two vectors lets us assume two vectors – a and b. The tail of b, as clearly seen here, coincides with the head of vector b. Then, the directed line segment will be from the tail of vector a to the head of vector b. The resulting vector can be considered as a+b. The vector addition has a clear similarity to the way forces and velocities add up.
Before we get into vector subtraction, we must first understand what a negative vector symbolizes. Let’s take a vector “a” and a negative vector “-a”. From the image, it is clear that the magnitude of both the vectors is the same, but the directions in which the vectors are facing are opposite. Now that we understand that, we can go on to define vector subtraction. Vector subtraction can be defined as the addition of two opposite vectors.
Vector multiplication has two primary methods present within it. One is the scalar product and the other is the vector product. The main difference between these two methods is the fact that we get a scalar value as a result through the first method, while the result obtained by using the second technique is also a vector in nature. For the sake of only knowing how to find the angle between two vectors, we will look at only the scalar product for now. The dot product entails taking the numeric coefficients of a particular vector, multiplying it with the numerical coefficient of the similar variable from the second vector, and finally adding together all the resultant product values.
Finding the angle
Now that we have the fundamental knowledge to find the angle θ between two vectors, let’s start with the formula for finding that angle’s cosine. The formula for finding the cosine between two angles is as follows:
The numerator in the above equation is the scalar product of both the vectors. The denominators a and b are both within the modular function, which determines the length of the particular vector (in this instance). The length of the vector can be obtained by squaring the respective coefficient present in the vector, adding them and then taking the square root of the resultant answer. Now, after the above operations and the fraction has been simplified, we will be left with a cosine function on the left side of the equation and a finite definite value on the right side of the equation. To find the angle theta, we need to perform one final operation, which entails taking the inverse cosine functions on both sides of the equations. This final step will give us the angle between the two vectors!