This presentation explains vectors and scalars, their methods of representation, their products and other basic things about vectors and scalars with examples and sample problems. This presentation is as per the course of DAE Electronics ELECT-212.

1. Scalar and Vector
IRFAN SULTAN
INSTRUCTOR (TELECOM.)
GOVT. COLLEGE OF TECHNOLOGY, PINDI-BHATTIAN
TECHNICAL EDUCATION AND VOCATIONAL TRAINING AUTHORITY (TEVTA)

2. Objectives
 Understand Vector Algebra
 Scalars and Vectors
 Unit Vector
 Vector addition and Subtraction
 Position and Distance Vectors
 Vector Multiplication
 Components of a Vector

3. Scalars and Vectors

4. Scalar
 Scalars are the quantities which only need Magnitude for their description.
 Examples:
 Time
 Mass
 Distance
 Temperature
 Electrical Potential

5. Vector
 Vectors are the quantities which need both, Magnitude and Direction, for their
description.
 Examples:
 Velocity
 Force
 Displacement
 Electric Field Intensity

6. Function
 If a quantity is dependent upon another quantity then it is called Function of that
independent quantity.
y = 2𝑥
𝑦 = 𝑥2 + 3𝑥 + 4
In both of these examples 𝑥 is independent quantity and 𝑦 is dependent upon 𝑥 and
𝑦 is called a function of 𝑥.

7. Field
 A Field is a function which is used to express a quantity in 3-dimensional (3D)
space.
𝑓(𝑥, 𝑦, 𝑧) = 3𝑥 + 3𝑦 + 3𝑧
𝑔 𝑥, 𝑦, 𝑧 = 𝑥3 + 2𝑦 + 𝑥𝑦
 There are two basic types of fields:
 Scalar Field
 Vector Field

8. Scalar Field
 A scalar field is a function which describe some specific value based on some
variables in 3-dimentional space.
Example:
𝑇(𝑥, 𝑦, 𝑧)
The function 𝑇 shows the temperature in a room at a point whose position is
described by 𝑥, 𝑦 and 𝑧.
Note that Temperature is a scalar quantity.
Hence scalar fields describe scalar quantities in 3D space.

9. Vector Field
 A vector field is a function which describe some specific value based on some
variables and a direction in 3-dimentional space.
Example:
𝑉(𝑥, 𝑦, 𝑧)
The function 𝑉 shows the velocity of air in a room at a point whose position is
described by 𝑥, 𝑦 and 𝑧.
Note that Velocity is a vector quantity.
Hence Vector fields describe vector quantities is 3D space.

10. Unit Vector

11. Presentation of a Vector
 A vector is shown with an arrow.
 The length of the arrow shows the magnitude of the vector.
 Direction of the arrow shows the direction of the vector.

12. Presentation of a vector
 Vectors are denoted with letters.
 A letter in bold-face represents a vector. E.g. A or a
 A small arrow at the top of a letter shows that the letter is denoting
a vector quantity
 ^ sign at the top of a letter is used to represent unit vectors.

13. Vector Example
 The vector in diagram
can also be written as:

14. Unit Vector
 A vector whose magnitude is One(1) is called a unit vector.
𝑉𝑒𝑐𝑡𝑜𝑟 = 𝑀𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑋 𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑉𝑒𝑐𝑡𝑜𝑟
𝑈𝑛𝑖𝑡 𝑉𝑒𝑐𝑡𝑜𝑟 = 1 𝑋 𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑉𝑒𝑐𝑡𝑜𝑟
𝑈𝑛𝑖𝑡 𝑉𝑒𝑐𝑡𝑜𝑟 = 1 𝑋 𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑉𝑒𝑐𝑡𝑜𝑟
 This equation shows that a unit vector is used to show the direction
of a vector.
Unit Means One

15. How to make a unit vector?
 If a vector is divide by its magnitude, the resultant vector would have a unit (1) magnitude and
would be in the direction of the vector, and hence would be called as the unit vector of that vector.

16. A unit vector in 3-Dimensions
 Magnitude of a vector is given by the following formula:
𝑨 = 𝐴𝑥2 + 𝐴𝑦2 + 𝐴𝑧2
 Hence the unit vector of A would be:
𝒂𝒙 =
𝑨
𝑨
=
𝐴𝑥𝒊 + 𝐴𝑦𝒋 + 𝐴𝑧𝒌
𝐴𝑥2 + 𝐴𝑦2 + 𝐴𝑧2

17. Activity

18. Vector Addition and
Subtraction

19. Vector Addition
 To or more vectors are added to get a third vector which is sum of the vectors.
 Two methods are used for addition of vectors:
 Head and Tail (Head-to-Tail) Rule
 Parallelogram Rule

20. Head and Tail Rule
 The method of vector addition in which head of
first vector is joined with the tail of 2nd vector
and the head of vector is joined with the tail of
third vector, and so on.
 The vector from the tail of 1st vector to the head
of last vector gives the sum of all the vectors and
is called Resultant vector.

21. Parallelogram Rule
 The method of vector addition in which a parallelogram is formed from the two
vectors which are to be added.

22. Vector Subtraction
 It is just like the addition of vectors. The only difference is that the vector(s) to be
subtracted is(are) reversed in direction and then added to the vector.

23. Some Simple Examples

24. Activity
If A = 2<30o and B = 1<60o
 Find:
 A+B
 A-B
 B-A

25. Assignment

26. Solution

27. Position Vector
and
Distance Vector

28. Position Vector
 A vector that gives the position of a point w.r.t origin is called the position
vector for that point.
Position vector in two dimensions
 A position vector for
a point P would be
written as:
𝒓𝒑 = 𝑂𝑃

29. Examples

30. Distance Vector
 A vector that gives the displacement between two points is called distance vector.
 Distance vector from point B to point A is written an:
𝒓𝐵𝐴 = 𝐵𝐴

31. Distance Vector

32. Example

33. Examples

34. Practice Exercise

35. Practice Exercise

36. Vector Multiplication

37. Concept of Vector Multiplication
 When two vector are multiplied together, the result is either:
 Vector
OR
 Scalar
 It gives rise to two types of Vector Multiplication
 Scalar Product (Dot Product)
 Vector Product (Cross Product)

38. Scalar (Dot) Product
 If the result of multiplication of two vectors is a scalar quantity, the product is called Scalar
Product. Since the sign of Dot (.) is used to denote this type of vector multiplication, it is also
called Dot Product.
 The formula for dot product of two vector A and B is:
𝐴. 𝐵 = 𝐴 𝐵 𝑐𝑜𝑠𝜃
OR
Where, Ax, Ay and Az are components of Vector A
And Bx, By and Bz are components of vector B.

39. Example

40. Properties of Dot Product
 If 𝑨 ⊥ 𝑩 then 𝑨. 𝑩 = 0
 If 𝑨 || 𝑩 then 𝑨. 𝑩 = 𝐴 |𝐵|
 Dot Product obeys Commutative Law.
 𝑨. 𝑩 = 𝑩. 𝑨
 Dot Product does not obey Associative Law.
 𝑨. 𝑩. 𝑪 ≠ (𝑨. 𝑩). 𝑪
 Scalar Multiplication
 c1A.c2B = c1c2 A.B
 Dot Product obeys Distributive Law.
 𝑨. 𝑩 + 𝑪 = 𝑨. 𝑩 + 𝑨. 𝑪
scalar multiplication property is
sometimes called the "associative
law for scalar and dot product"

41. Properties of Dot Product
 Dot Product of a Vector with itself equals the square of magnitude of that vector.
 𝑨. 𝑨 = |𝑨|2 = 𝐴2
 For any unit vector:
 𝒂𝒙. 𝒂𝒙 = 1
 𝒂𝒚. 𝒂𝒚 = 1
 𝒂𝒛. 𝒂𝒛 = 1
 𝒂𝒙. 𝒂𝒚 = 0
 𝒂𝒚. 𝒂𝒛 = 0
 𝒂𝒛. 𝒂𝒙 = 0

42. Applications of Dot Product
 Generally Dot Product is used:
 To find angle between two vectors.
 To find components of a vector in a specific direction.
 To find Work Done due to a constant force.

43. Vector(Cross) Product
 If the result of multiplication of two vectors is a vector quantity, the product is called Vector
Product. Since the sign of Cross (X) is used to denote this type of vector multiplication, it is also
called Cross Product.
 The formula for cross product of two vector A and B is:
𝐴. 𝐵 = 𝐴 𝐵 𝑠𝑖𝑛𝜃 𝑛
OR
𝐴 × 𝐵 =
𝑖 𝑗 𝑘
𝐴𝑥 𝐴𝑦 𝐴𝑧
𝐵𝑥 𝐵𝑦 𝐵𝑧
Where, Ax, Ay and Az are components of Vector A
And Bx, By and Bz are components of vector B.