6.2 Geometric introduction to vectors

. Geometric introduction to vectors A vector v  is represented as a directed straight line segment in a -dimensional space  , with an initial point A = ( a a a ∈  and an end point B = ( b b b ∈  , and it is denoted by AB . The length of the line segment AB is the magnitude of the vector v  and the direction from A to B is the direction of the vector v  . Hereafter, a vector will be interchangeably denoted by v  or AB . Two vectors AB and CD in  are said to be equal if and only if the length AB is equal to the length CD and the direction from A to B is parallel to the direction from C to D . If AB and CD are equal, we write AB CD , and CD is called a translate of AB It is easy to observe that every vector AB can be translated to anywhere in  , equal to a vector with initial point U ∈  and end point V ∈  such that AB = UV . In particular, if O is the origin of  , then a point P ∈  can be found such that AB = OP . The vector OP is called the position vector of the point P . Moreover, we observe that given any vector v  , there exists a unique point P ∈  such that the position vector OP of P is equal to v  . A vector AB is said to be the zero vector if the initial point A is the same as the end point B . We use the standard notations ˆ ˆ , , i j k and to denote the position vectors of the points ( , , ),( , , ),( , , ), and ( , , ), respectively. For a given point , a a a ∈  , ˆ a i a j a k is called the position vector of the point ), a a a