y and z -axes respectively, are called its direction angles . The cosine values of these angles, i.e., cos α , cos β and cos γ are called direction cosines of the vector , and usually denoted by l , m and n , respectively. Fig . From Fig .
, one may note that the triangle OAP is right angled, and in it, we have . Similarly, from the right angled triangles OBP and OCP, we may write cos and cos y z β = γ = . 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 , and denoted as a , b and c , respectively.
Fig . A Note One may note that l + m + n = but a + b + c ≠ , in general. . Types of Vectors Zero Vector A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as .
Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors represent the zero vector, Unit Vector A vector whose magnitude is unity (i.e., unit) is called a unit vector. The unit vector in the direction of a given vector is denoted by ˆ a .
Coinitial Vectors Two or more vectors having the same initial point are called coinitial vectors. 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 Negative of a Vector A vector whose magnitude is the same as that of a given vector (say, ), but direction is opposite