Chapter one
The Vectors
Chapter one
The vectors
Scalar quantity: the quantity that is completely
1-1 physical Quantities
specified by a number and appropriate units,
such as Temperature, mass, time,………etc.
Vector quantity: is a physical quantity that is
completely specified by a number and appropriate
units plus a direction in which it is affect, such as
displacement, velocity, acceleration, force,….etc.
r
Let A a vector, it's magnitude A , and n̂ is the unit vector defined as a
dimensionless vector having a magnitude of exactly (1). Then we can write
r
the vector A as:
r
r
A = n̂ A
A = n̂ A
1-2 Multiplying a Vector by a Scalar
r
r
If A a vector multiplied by a scalar quantity (n) then (n A ) is the vector
r
has the same direction of A and magnitude (n A).
Chapter one
The Vectors
1-3 Adding Vectors
r
r
Draw vector A , then draw vector B write it's tail starting from the tip of
r
r
r
r
r
B . The vector C is the resultant of adding A and B . The resultant C is
r
r
the vector drawn from the tail of A to the tip of B .
r r r
C = A+B
r
B
r
A
r
A
r
B
*Properties of Adding
r
B
(a) The commutative property
r r r r
A + B= B + A
(b) Associative Property
r r
r r
r r
(A + B) + C = A + (B + C)
r
A
r
C
r
r r
A + ( B + C)
r
A
r r
B+C
r
B
1-4 Subtracting Vectors
r
r
For two vector A and B
r r r
r r r r
r
A - B = A + (-B) , where A - B ¹ B - A
r r
A-B
r
- rB
A
r
B
Chapter one
The Vectors
1-5 Vectors Analysis
We use the symbols i , j and k to represent unit vectors in the position x, y
and z direction respectively. The magnitude of each unit vector equals
unity, that is
i = j = k =1
r
In xy-plane , we represent the vector A as:
A y ˆj
r
A = A î + A ˆj
x
r
A
y
A x î
r r
For A , B are two vectors, where
r
r
A = A x î + A y ˆj , B = B x î + B y ˆj
r
r
The resultant of A and B is R
r r
R= A + B =( A x î + A y ˆj ) + ( B x î + B y ĵ )
R= (Ax+Bx) î +(Ay+By) ˆj
Then , the component of the resultant R are:
Rx=Ax+By
, Ry=Ay+By
Then the magnitude of the resultant R and it's direction are written as:
R= R 2x + R 2y
= (A x + B x ) 2 + (A y + B y )
2
And
R
or q = tan -1 y
Rx
Rx
r
r
Now, for A and B are two vectors have x , y and z components, we
tan q =
Ry
express these vectors in the form :
r
A = A x î + A y ˆj + A z k̂
r
B = B x î + B y ĵ + B z k̂
Chapter one
And the resultant R of the vectors is:
r r
R= A + B = (A x + B x )î + (A y + B y )ˆj + (A z + B z )k̂
The Vectors