Mathematical Physics Vol 1

1.4 Operations on vectors

31

If the origin is chosen for the beginning of the vector, then the coordinates of the end point of the vector are equal to the measures of the vector. It has already been said that a vector constructed in this way is called position vector and is usually denoted by r . It can also be observed from (1.37) that the measures a x , a y , a z of the vector a do not depend on the choice of the start point of a , because if the vector a is moved along the direction AB , then the coordinates of the points A and B change for the same value, and their difference remains the same. Thus, if a fixed Cartesian coordinate system is given, then each vector is uniquely determined by an ordered triple of numbers (coordinates). The zero vector 0 can defined accordingly as a vector whose coordinates are [ 0 , 0 , 0 ] . It is said that two vectors a = [ a x , a y , a z ] and b = [ b x , b y , b z ] are equal iff their respective coordinates are equal. Namely, the vector equation: a = b (1.40) is equivalent to three scalar equations: a x = b x , a y = b y , a z = b z . (1.41) The following operations can now be defined using the coordinates: Sum of vectors (Fig. 1.21): c = a + b =[ a x + b x , a y + b y , a z + b z ]=[ c x , c y , c z ] (1.42)

y

b

b y

c y

c

a y a

a x

b x

x

c x

Figure 1.21: Sum of vectors expressed in terms of their projections (components).

Multiplication of a vector by a scalar

α a =[ α a x , α a y , α a z ]

(1.43)

Scalar product

a · b = a x b x + a y b y + a z b z

(1.44)

given that

e i · e j = δ i j =

1 , i = j 0 , i̸ = j .

(1.45)