Lecture (2) Vector Addition , Subtraction, Multiplication and Division

Vector Addition and Subtraction
Consider that the vector A has three components A x, A y and A z in the x , y and z directions, respectively. According to Fig.1.6, the vectors A x a x, A y a y and A z a z are the components of the vector A in the x, y and z directions, respectively. The resultant of these vectors is
A= A x a x + A y a y + A z a z (1- 16)
Similarly, vectors B and C can be expressed as
B= B x a x + B y a y + B z a z (1- 17)
C= C x a x + C y a y + C z a z (1- 18)
Z
A
Az az
Y
Ax ax
X Ay ay

Fig.( 1- 6) : Three component vector
The magnitude of the vector A can be written as
A= |A|=?(A_X^2+ A_y^2+ A_Z^2 ) (1 – 19 )
The unit vector in the direction of the vector A is

a_A= A/|A| = (A_X a_X+ A_y a_y+ A_Z a_Z)/?(A_X^2+A_y^2+A_Z^2 ) ( 1 – 20 )
Vector addition can be obtained by parallelogram and nose – to – tail or head to tail rules . Two vectors A and B started from the same point as shown in Fig. (1 – 7 ) . The resultant vector can be calculated as
R a = A + B = B + A (1 – 21 )
Substituting Eqs. ( 1 – 16 ) and ( 1 – 17 ) in to Eq. (1 - 21 )yields
R a = (A x + Bx )a x + ( A y + B y ) a y + ( A z + B z ) a z (1 – 22 )
B B B A
Ra Ra
A A Ra
Fig. (1 – 7 ) : Parallelogram and head – to – tial rules

Vector subtraction is defined as the special addition of two vectors. Consider two vectors B and C for vector subtraction as shown in Fig.(1- 8). The vector subtraction can be represented as
R S = B – C = B + ( - C ) ( 1 – 23 )
Where
- C is the negative vector of C. Substituting Eqs.( 1 – 17 ) and ( 1 – 18 ) in to Eq. ( 1 – 23 ) provides
R S = ( B x a x + B y a y + B z a z ) + ( - C x a x - C y a y - C z a z ) ( 1 - 24 )
RS = ( BX – CX ) aX + ( By - C y ) ay + ( BZ – C Z ) aZ