6.2 Geometric introduction to vectors · Part 2

which is the directed straight line segment with initial point ( , , ) and end point a a a . All real numbers are called scalars. Fig. . U V B P O C D Vector - - Applications of Vector Algebra Given a vector AB , the length of the vector is calculated by where A is a a a and B is ( , ). b b b In particular, if a vector is the position vector b of ( , ), b b b then its length is . A vector having length is called a unit vector . We use the notation ˆ u , for a unit vector. Note that ˆ ˆ , i j , and ˆ k are unit vectors and is the unique vector with length . The direction of is specified according to the context. The addition and scalar multiplication on vectors in -dimensional space are defined by ˆ b i b k α  = ˆ a i a k ; where a  = a i a j a k b i b j b k ∈  and α ∈  . To see the geometric interpretation of a , let a  and b , denote the position vectors of a a a ( , B b b b , respectively. Translate the position vector b to the vector with initial point as A and end point as ( , C c c c , for a suitable ( , c c c ∈  . See the Fig ( . ). Then, the position vector c  of the point ( , c c c is equal to a The vector a α  is another vector parallel to a  and its length is magnified (if ) α > or contracted (if ) < < . If α < , then α  is a vector whose magnitude is | α times that of a  and direction opposite to that of  a . In particular,