1. General Physics 101 PHYS Dr. Zyad Ahmed Tawfik
Website : zyadinaya.wordpress.com

2. Lecture No.5
SUMMARY
On vectors
Vectors

3. All Physical quantities can be divided into two types:
1) Scalars are : A physical quantity which requires only magnitude for their complete description.
-For example : mass, speed, energy, work, temperature and pressure.
2) Vectors are. A Physical quantity which requires both the magnitude and direction for their complete description.
-For example: Weight, velocity, force and displacement .

4. Addition of Vectors

5. Adding vector There are three methods to adding Vector
1- Graphical or called (Geometrical Method)
2- Pythagorean Theorem
3- Analytical Method or called Component's Method

6. Geometric Method

7. 1) Add vectors A and B graphically by drawing them together in a head to tail arrangement.
2) Draw vector A first, and then draw vector B such that its tail is on the head of vector A.
3) Then draw the sum, or resultant vector, by drawing a vector from the tail of A to the head of B.
4) Measure the magnitude and direction of the resultant vector

8. RESULTANT VECTOR is the is the vector that 'results' from adding two or more vectors together.
RESULTANT VECTOR = R
Vector 2= B
Vector 1 = A

9. In this method To add vectors, we put the start point of the second vector on the end point of the first vector. The resultant vector is distance between start point and end point .
To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).
End point
Start point

10. Pythagorean Theorem

11. The Pythagorean Theorem
The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

13. Component method

14. In this method Each vector has two components :
the x-component and the y-component
If the vectors are in secondary directions :
(NW, NE, SW or SE directions)
Ax = A cos θx
Ay = A sin θx
where:
A = the given vector value
θx = the given angle from x -axis
Ax = the x – component of vector A
Ay = y – component of vector A

15. and Bx = B cos θ and By = B sin θ
To calculate the resultant vector by component method.
-for example vector A and vector B :
1- calculate the component vectors for vector A and B
so : Ax = Acos θ and Ay = A sin θ
and Bx = B cos θ and By = B sin θ
2- We can get Rx =Ax+ Bx and Ry = Ay + By
3- Calculate the magnitude of the resultant by the
Pythagorean theorem .
4-Determine the angle θ by the equation

17. Vectors Components
The figure shows a vector A and an xy-coordinate system.
We can define two new vectors parallel to the x and y axes, named the component vectors of A,
Slide 3-30

18. NEED A VALUE OF ANGLE!
To find a numeric value for the angle, we used the following laws.
Hypotenuse
Opposite q
Adjacent

19. Also: cos θ = Ax/A Or Ax = A cos θ.
&
sin θ = Ay/A Or Ay = A sin θ Ay θ Ax

20. Unit Vector Notation

21. Unit Vector Notation,
A unit vector is a vector that has a magnitude of one unit and can have any direction.
Traditionally i^ (read “i hat”) is the unit vector in the x direction
and j^ (read “j hat”) is the unit vector in the y direction. |i^|=1 and |j^|=1, this in two dimensions ,
and motion in three dimensions with ˆk (“k hat”) as the unit vector in the z direction

22. If A & B are two vectors, where
A = axi + ayj + azk& B = bxi + byj + bzk Then the:
magnitude of A+B in this case is
magnitude of A-B in this case is
Example 1
Tow vector A = 3i - 6j -5k & B = 2i + 3j - 2k
Find the magnitude of A+B and A-B ?
Answer
First the magnitude of A+B we will use the last equation so

23. Secondly the magnitude of A-B we will use the last equation so

24. Product of Vectors

25. There are two kinds of vector product :
The first one is called scalar product or dot product because the result of the product is a scalar quantity. 
The second is called vector product or cross product because the result is a vector perpendicular to the plane of the two vectors.

26. Vector product or cross product
If A & B are vectors,
their Vector (Cross) Product is defined as:
AxB is read as “A cross B”
A x B= AB sinθ
where θ is the angle between A & B
Dot product
If A & B are vectors,
their Dot Product is defined as:
A.B is read as “A dot B”
A.B= AB Cosθ
where θ is the angle between A & B

27. Case 1, (With out angle θ) AB = AxBx + AyBy + AzBz
In dot product
Case 1, (With out angle θ)
for example If A & B are two vectors, where
A = Axi + Ayj + Azk & B = Bxi + Byj + Bzk
Then, their Scalar Product is defined as:
AB = AxBx + AyBy + AzBz
where

28. If A & B are two vectors, and θ is the angle between them,
Case 2, (With angle θ)
If A & B are two vectors, and θ is the angle between them,
Then, their Scalar Product is defined as:
AB = AB cos θ

29. How can you calculate the angle between tow vector A and B if A = axi + ayj + azk, B = bxi + byj + bzk by using dot product ?
Answer 1- first calculate dot product A . B = ax bx |ay by| az bz 2- calculate the magnitude A and the magnitude B Where magnitude 3- using equation AB = AB cos θ to find the angel θ between vector A and vector B by

30. Some important low in this chapter

31. and A.B= AB Cosθ Ax = Acos θ & Ay = A sin θ b) - With angle θ
1- To calculate The magnitude A+B
a) - With θ=90 so Cos θ= Cos90=0
b) - With angle θ
2) To calculate The components and magnitude of vector
for example the components of vector A are
Ax = Acos θ & Ay = A sin θ
3) To calculate the resultant vector by component method
And the magnitude A is
and
and
4) Low of Dot product
and
A.B= AB Cosθ

32. A.B= AB Cosθ and A x B= AB sinθ
4) Low of Dot product
A.B= AB Cosθ
and
6). if A & B are two vectors, where A = axi + ayj + azk& B = bxi + byj + bzk 1-Then the magnitude of A+B and A-B are And
A x B= AB sinθ
5) Low of Cross product

33. Thank You for your Attention