#### description

- Position Vector
- Direction Cosines and Direction Ratios of a Vector

#### definition

A quantity that has magnitude as well as direction is called a vector.

A directed line segment is a vector in above fig., denoted as `vec (AB)` or simply as `vec a` , and and read as 'vector `vec (AB)`' or 'vector `veca`'.

The point A from where the vector `vec (AB)` starts is called its initial point, and the point B where it ends is called its terminal point. The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as

|`vec (AB)`|, or |`vec a`|, or a. The arrow indicates the direction of the vector.

#### notes

Let ‘l’ be any straight line in plane or three dimensional space. This line can be given two directions by means of arrowheads. A line with one of these directions prescribed is called a directed line in following first and second fig .**Position Vector:**

The three dimensional right handed rectangular coordinate system in following first fig.

Consider a point P in space, having coordinates (x, y, z) with respect to the origin O(0, 0, 0). Then, the vector having O and P as its initial and terminal points, respectively, is called the position vector of the point P with respect to O.

Using distance formula `|vec (OP)| = sqrt(x^2 + y^2 + z^2)`

In practice, the position vectors of points A, B, C, etc., with respect to the origin O are denoted by `vec a , vec b , vec c` etc., respectively in above second fig.

**Direction Cosines:**

The position vector `vec (OP) (or vec r)` of a point P(x, y, z) as in following Fig.

The angles α, β, γ made by the vector `vec r` with the positive directions of x, y and z-axes respectively, are called its direction angles.

The cosine values of these angles, i.e., cosα, cosβ and cosγ are called direction cosines of the vector `vec r` , and usually denoted by l, m and n, respectively.

one may note that the triangle OAP is right angled, and in it, we have `cos alpha = x/r`( r stands for |`vec r`|). Similarly, from the right angled triangles OBP and OCP, we may write `cos beta= y/r` and `cos gamma = z/r`. Thus, the coordinates of the point P may also be expressed as (lr, mr,nr). The numbers lr, mr and nr, proportional to the direction cosines are called as direction ratios of vector `vec r` , and denoted as a, b and c, respectively.