Vector Algebra -

Last Updated on: 15th February 2024, 12:51 pm

Vector Algebra

Vectors

We frequently come across with scalar values like as length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance etc. In science, we also come across vector quantities like displacement, velocity, acceleration, force, weight, momentum, electric field intensity which have to be measured in different way. In this chapter we will discuss about the combination of these two concepts in Vector Algebra.

In the above figure, the first two represent direction, in the third, the definite section ‘ab’ represents a scalar value l and a direction. The combination of scalar value and a direction is called a vector.

A Quantity having magnitude as well as direction is called a vector. So, a line with a definite segment, known as vector, is represented as $\displaystyle \overrightarrow{{AB}}$ .

Initial Point of Vector: The starting point (A) of the vector is called Initial Point
Terminal Point Of Vector : The end point (B) of the vector is called terminal point.
Magnitude of Vector: The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as $\displaystyle \left| {\overrightarrow{{AB}}} \right|$ or $\displaystyle \left| {\overrightarrow{a}} \right|$ , or $\displaystyle \overrightarrow{a}$ .

Types of Vectors

Zero Vector : A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as $\displaystyle \overrightarrow{0}$ . Zero vector has zero magnitude and indeterminate direction (considered as having any direction). The vectors AA, BB, represent zero vector,
Unit Vector : A vector of magnitude 1 unit is called a unit vector. The unit vector in the direction of a given vector $\displaystyle \overrightarrow{a}$ is denoted by $\displaystyle \widehat{a}$ . Two unit vector may not be equal, though magnitude is same (1 unit), if direction is different.
Position Vector : If O is taken as reference origin and A is an arbitrary point in space then the vector $\displaystyle \overrightarrow{{OA}}$ is called as the position vector of the point. Position vector just denotes the position or location of a point in the three-dimensional Cartesian system with respect to origin.
Co-initial Vectors : Two or more vectors having the same initial point are called coinitial vectors (e.g. vector $\displaystyle \overrightarrow{{AB}}$ , $\displaystyle \overrightarrow{{AC}}$ ).
Like & Unlike Vectors : Vectors having the same direction (irrespective of magnitude) are known as Like vectors, while having the opposite direction to each other are called Unlike vectors
Co-planar Vectors : Three or more vectors lying in the same plane or parallel to the same plane are known as Co-planar vectors
Displacement Vector : If a point is displaced from position A to B then the displacement AB represents a vector $\displaystyle \overrightarrow{{AB}}$ known as the displacement vector.
Collinear Vectors : Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
Equal Vectors : Two vectors are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as $\displaystyle \overrightarrow{a}=\overrightarrow{b}$ .
Negative of a Vector A vector whose magnitude is the same as that of a given vector, but direction is opposite to that of it, is called negative of the given vector. For example, if vector $\displaystyle \overline{{AB}}$ is negative of the vector $\displaystyle \overline{{BA}}$ and written as $\displaystyle \overline{{AB}}=-\overline{{BA}}$