Vector analysis

Vector analysis

The various quantities of any field can be classified as, Scalar quantity and Vector quantity. In this article, we will discuss the basics of vector analysis.

Scalar

A quantity that has only magnitude (value). To represent this type of quantity, the direction is not necessary. The value of this quantity may be positive or negative.

Examples: temperature, mass, volume, speed, electric charge, etc.

Vector

A quantity that has both magnitudes and the specific direction in the space is known as a vector quantity. These quantities define in two dimensional, three dimensional or n-dimensional spaces.

Examples: force, current, velocity, displacement, acceleration, electric field intensity, magnetic field intensity, etc.

Representation of Scalar and Vector

A single line represents a scalar. The length of this line gives the magnitude of that quantity. Therefore, the direction is not required to represent the scalar quantity.

A single line with an arrow represents a vector. The length of the line gives the magnitude, and the arrowhead shows the direction of that vector.

Vector has a start point and endpoint. In below figure, vector \overrightarrow{OA} is denoted. The starting point is O, and the terminating point is A. length of this vector is P. So, the distance between the point O and A is P.

Scalar and Vector

Unit Vector

The function of the unit vector is to indicate the direction of the vector. The magnitude of the unit vector is always one (unity). In vector analysis, unit vector plays an important role.

To indicate the direction of the vector, a unit vector is used. In the above figure represents vector \overrightarrow{OA}.

The unit vector represent by \overrightarrow{a} or \hat{a}. In this figure, the unit vector in the direction of vector \overrightarrow{OA} is \overrightarrow{a}_O_A or \hat{a}_O_A.

Thus, vector \overrightarrow{OA} completely represented by magnitude P and unit vector  \overrightarrow{a}_O_A.

(1)   \begin{equation*} \begin{align} \overrightarrow{OA} &= |\overrightarrow{OA}| \overrightarrow{a}_O_A \\ \overrightarrow{OA} &= P \: \overrightarrow{a}_O_A \\ \overrightarrow{a}_O_A &= \frac{\overrightarrow{OA}}{P} \end{align} \end{equation*}

The mathematical operations such as addition, subtraction and multiplication can be performed with the vectors. In this vector analysis, here we will discuss addition and subtraction.

Addition of Vectors

Consider two vectors as shown below figure. We want to find the sum of these two vectors. Let us consider, the first vector is \overrightarrow{A} and the second vector is \overrightarrow{B}.

The procedure of adding these vectors is known as the parallelogram method. In this method, make a parallelogram by drawing parallel vectors.

Addition of vectors

In this case, make a parallel vector of \overrightarrow{A} and put it at the tip of vector \overrightarrow{B}. Similarly, make a parallel vector \overrightarrow{B} and put it at the tip of vector \overrightarrow{A}.

Now, connects the intersecting point with the origin O and make a vector \overrightarrow{C}. This vector C is the sum of vector \overrightarrow{A} and \overrightarrow{B}.

(2)   \begin{equation*} \begin{align} \overrightarrow{C} &= \overrightarrow{A} + \overrightarrow{B} \\ Cumulative\: law: \overrightarrow{A} + \overrightarrow{B} &= \overrightarrow{B} + \overrightarrow{A}\\ Associative\: law: \overrightarrow{A} + (\overrightarrow{B} +\overrightarrow{C}) &= (\overrightarrow{A} + \overrightarrow{B}) + \overrightarrow{C} \\ Distributive\: law: \alpha (\overrightarrow{A} + \overrightarrow{B}) &= \alpha \overrightarrow{A} + \alpha \overrightarrow{B} \end{align} \end{equation*}

Subtraction of Vectors

Subtraction of vectors can be obtained from the rule of addition. Suppose, if you want to subtract \overrightarrow{B} from \overrightarrow{A}. Now based on the addition rule, it can be represents as

(3)   \begin{equation*} \overrightarrow{C} = \overrightarrow{A} + (\overrightarrow{-B}) \end{equation*}

The reverse sign of \overrightarrow{-B} means it reverses the direction and then adds it to \overrightarrow{A}. This is shown below the figure.

Subtraction of vectors

Identical Vectors

Two vectors are said to be identical if their difference is zero. Thus, \overrightarrow{A} and \overrightarrow{B} are identical if \overrightarrow{A} - \overrightarrow{B} = 0. Such vectors are called equal vectors.

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