Geometrical Representation, Position Vector and Rectangular Coordinate System

Geometrical Representation of a Vector

Geometrically a vector is represented by a line segment with an arrow head at its one end (the ray). The length of the line segment describes the magnitude and the arrow head indicates the direction of the vector.

In the figure, the end A of the ray AB is called the tail or the initial point of the vector and the end B is called the terminal point with the arrow head denoted by $\overrightarrow{AB}$

.

Usually, the vectors are denoted by bold face letters $a,b,c$$\pmb{a}, \pmb{b}, \pmb{c}$ or $\vec{a}, \vec{b}, \vec{c}$

,b

,c

etc.

Position Vector

The vector used to specify the position of a point P with respect to origin O is called position vector of P. The tail of this vector is at origin and head at the point P. Thus $\overrightarrow{OP}$

is the position vector of point P with respect to O.

Result:

If $\vec{a}$

and $\vec{b}$

are position vectors of points A and B respectively, then $\overrightarrow{AB} = \vec{b} - \vec{a}$

=b

−a

.

Proof:

If $\vec{a}$

and $\vec{b}$

are position vectors of points A and B respectively, then:

Using triangle law of addition, we have:

Rectangular or Cartesian Coordinate System

A rectangular or Cartesian coordinate system consists of two number lines which are mutually perpendicular.

Their point of intersection is called the origin, usually denoted by the letter 'O'. Horizontal line is called x-axis and vertical line is called y-axis. Ordered pairs of numbers can be represented on a Cartesian plane, which is sometimes called the Coordinate plane or xy-plane. Each point in the plane corresponds to an ordered pair $(x,y)$ of real numbers $x$ and $y$, called the coordinates of the point.

Cartesian plane has got this name after the French mathematician Rene Descartes (1596 - 1650). He developed the idea of the coordinate plane and the correspondence of ordered pairs of numbers to points in the coordinate plane. Each point in the plane can corresponds to only one ordered pair and vice versa.

Let R be the set of real numbers, then the Cartesian plane is defined as:

Any point $\mathbf{P}(x,y)$ is uniquely determined by its coordinates $x$ and $y$. For a given vector $\vec{u}$

in the plane, there exists a unique point $\mathbf{P}(x,y)$ in the plane such that: