A quantity that has both direction and magnitude is called a vector. Vectors are essential while dealing with many real-life situations, such as situations involving velocity or force. Let us consider the forces operating on a boat when crossing a river. The motor of a boat generates a force in one direction, whereas the current flow of the river will generate a force in another direction. Here, both the forces are treated as vectors. We have to take both the direction and magnitude of each force into consideration if we want to know the direction where the boat will go. We can say that; vector algebra is a must in such circumstances of operations on forces because we can not combine them like numbers with only magnitude and no direction.
Vector algebra or vector analysis can generate simplified proofs of geometry is another aspect of vector algebra. One of the critical operatives to study and memorize in vector algebra will be the vector dot product. The importance of vector algebra is multifold. Vector algebra has developed from a requirement to simplify and interpret elements. If we examine the classical geometry books, we could trace the progression of vector algebra. Initially, there was something describing the algebra of segments. It gives a direct application to displacements and their addition. Also, it is essential to both physics and Mathematics. Below listed points reveal the importance of vector algebra in different areas of operations.
- Geometric transformations and related conversions become more comfortable to handle.
- The first and foremost thing is that we can gain instantaneous insight into Geometry.
- Some of the problems in analytical coordinate geometry become easy to simplify and solve.
- Concepts that we acquired in this algebra are essential to understand the several concepts of linear algebra.
- In Physics, vector algebra helps describe the planar and 3D movement of objects and calculating trajectories.
- Vector algebra also plays a crucial role in particle mechanics and fluid mechanics.
- Vectors are used widely in electromagnetism.
- Instead of vectors, Maxwell had practised quaternions, and Heaviside provided us with the latest form of the equations which are far more comfortable to handle.
Apart from the above, vector algebra procedures are used as often as algebra in math and physics. However, the concepts of dot and cross products are useful in solving two and three-dimensional geometric problems. Now, let’s understand the answer for the question; what is the vector dot product used for? The answer will be: the dot product is used to define lengths and angles. Here, the angle means the cosine of the angle of two vectors is the ratio of their dot product and the product of their sizes (i.e., lengths).
When coming to the cross product of three vectors, which allows us to estimate a vector that is orthogonal to two given vectors. For example, calculating torque involves the cross products. Vector algebra is a further generalization of algebra to vectors and applied to resolving the linear equations used to the neural network to reflect artificial neural networks’ operation.