In mathematics, a vector is a quantity which has both magnitude and direction. Two operations are defined on vectors in maths, and these both have a straightforward geometric representation. We can draw a vector as a line segment with an arrow at one of its ends. The end without an arrow is called the ‘start’ point of the vector, and the other end with an arrow is called ‘end’ of the vector. Also, the length of the line segment is called the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is considered from its tail to its head.
Historically speaking, vectors were introduced in fields like geometry and others (for example in mechanics) which is mostly before the implementation of the idea of vector space. Therefore, one addresses often of vectors without even specifying its actual name to which they belong. Notably, in a Euclidean space, one interprets spatial vectors; these are also called Euclidean vectors. These used to denote such quantities which have both direction and magnitude. This may be added and estimated, that means when multiplied by a real number to form a vector space.
A vector or vector quantity gives not only the information about the magnitude but also the direction of the quantity. When providing instructions to a building, it is not sufficient to say that it is nine miles away. Still, the direction of those nine miles is required to provide for the data to be helpful for further procedure. Variables that are vectors will be represented or denoted with a boldface variable. However, it is common that vectors appear by denoting small arrows above the variable. When coming to the properties of vectors, the following points can be observed.
- The magnitude of a vector is always a positive number, in other words, it is the absolute value (ignore negative sign) of the length of the vector
- Although the quantity may not be a length, it may be a force, acceleration, velocity and so on
- A negative sign in front of a vector does not indicate any change in the magnitude, but rather with respect to the direction of the vector
In the above example, distance is the scalar quantity (nine miles). Still, displacement is the vector quantity (nine miles to the southeast). In the same manner, speed is a scalar quantity, whereas velocity is a vector quantity. Besides, there are different types of vectors, namely, unit vector, basis vector, null vector or zero vector. The unit vector ‘a’ is generally written with a carat and is read as “a-hat”. The reason behind this is, a hat on the variable is visible as the carat looks. Various theorems defined on vectors such as stokes theorem in relation to calculus and the fundamental theorem of vectors, etc.,
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In calculus, vectors play an important role when it comes to higher-level mathematics. Several theorems and formulas exist in differential geometry in order to solve many problems. We can also extend the applications of vectors to form more complicated mathematical structures. Alternatively, we might think of vectors as subsets of these structures. More precisely, the elements of the vector necessarily have a mathematical structure named a field.