
A vector that has no specific position in space is called a free vector.
Consider the position of a point A relative to a fixed origin O.� OA is called the position vector of A relative to O.� This displacement is unique and cannot be represented by another line.
Note:� Any vector AB can be written in terms of the position vectors of A and B:
Let a be a nonzero vector and k be any real number. We define a new vector, ka, as follows.
(a) If k = 0, the ka = 0.
(b) If k > 0, the ka has the same direction as a and ka = ka.
(c) If k < 0, the ka is in the opposite direction to that of a and ka = ka.
For all vectors a and b and all scalars l and m, we have:
(a) l(ma) = (lm)a,
(b) (l � m)a = la � ma,
(c) l(a � b) = la � lb.
(a)� A unit vector is a vector with a magnitude of one unit.� A unit vector in the direction of a will be typed as .
(b)� Any vector r can be expressed as the product of its magnitude and the unit vector in the same direction: r = rr^
(c) Three important unit vectors are defined as follows:
�
If a point P divides a line segment AB in the ratio m : n, then�
�� 
Note:� If M is the midpoint of AB, then OM = (OA + OB)/2.
If P(a, b, c) is any point in 3D Cartesian space, then the position vector of the point P is OP = p where
�
�  a  �  
(a) p = ai + bj + ck or�  �  b  � 
�  c  � 
(b) p = (a^{2} + b^{2} + c^{2})^{1/2}
= d
�
�  a/d  �  
(c) p^ = p/p =�  �  b/d  � 
�  c/d  �� 
A��� Parallel
Let a and b be nonzero vectors.�
Then a is parallel to b if and only if 
B��� Collinear
Three points A, B and C are collinear if and only
if

C��� Coplanar
Let a and b be nonzero, nonparallel vectors.
A vector c is coplanar with a and b if and only if 