Chapter W.R. Hamilton ( - ) Now observe that if we restrict the line l to the line segment AB, then a magnitude is prescribed on the line l with one of the two directions, so that we obtain a directed line segment (Fig . (iii)). Thus, a directed line segment has magnitude as well as direction.
Definition A quantity that has magnitude as well as direction is called a vector. Notice that a directed line segment is a vector (Fig . (iii)), denoted as or simply as , and read as ‘vector ’ or ‘vector ’. The point A from where the vector 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 | |, or | |, or a . The arrow indicates the direction of the vector. A Note Since the length is never negative, the notation | | < has no meaning. Position Vector From Class XI, recall the three dimensional right handed rectangular coordinate system (Fig .
(i)). Consider a point P in space, having coordinates ( x , y , z ) with respect to the origin O( , , ). 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 (from Class XI), the magnitude of (or ) is given by | |= x y z In practice, the position vectors of points A, B, C, etc., with respect to the origin O are denoted by , , , etc., respectively (Fig .
(ii)). Fig . A O P ° X Y Z X A O B P( x,y,z C P( x,y,z x y z Direction Cosines Consider the position vector of a point P( x , y , z ) as in Fig . .
The angles α , β , γ made by the vector with the positive directions of x ,