1. Chapter 3
VECTORS

2. OUTLINE OF CHAPTER 3: VECTORS
Introduction
Addition of Vectors
Subtraction of Vectors
Scalar Multiplication of Vectors
PARALLEL VECTORS
Components of Vectors IN TERMS OF UNIT VECTORS
POSITION VECTORS
Components of Vectors
Magnitude of Vectors
UNIT VECTORS
OPERATIONS OF VECTORS BY COMPONENT
Product of 2 Vectors
Application of Scalar/Dot Product & Cross Product

3. VECTORS SCALARS 3.1 introduction Has magnitude only
Has magnitude (represent by length of arrow)
Direction (direction of arrow either to the right, left, etc)
E.g: displacement, velocity, acceleration, force.
- move the brick 5m to the
right,
SCALARS
Has magnitude only
No direction
E.g: mass, temperature, length, area.
- move the brick 5m

4. 3.2 vector representation
Use an arrow connecting an initial point A to terminal point B
Written as a, w, or u.
Magnitude of
B (terminal point)
B (initial point)
A (initial point)
A (terminal point)

5. 3.3 vector negative
The vector −a would represent the vector in opposite direction, but has the same magnitude as
vector a. Geometrically, if B B A A

6. 3.4 two equal vectors
Two vectors a and b are said to be equal if and only if they have the same magnitude and direction.

7. 3.5 addition of vectors (i) The Triangle Law
Any 2 vectors can be added by joining the initial point of b to the terminal point of a.
Example 3.1:

8. Exercise

9. (ii) The Parallelogram Law
If 2 vector quantities are represented by 2 adjacent sides of a parallelogram, then the diagonal of parallel will be equal to the summation of these 2 vectors.
Example 3.2:
*The parallelogram law is affected by the triangle law.

10. (iii) The Sum of a Number of Vectors
The sum of vectors is given by the single vector joining the start of the first vector to the end of the last vector.
Example 3.3:
*Note: The addition of vector is satisfied the commutative (a+b = b+a)

11. 3.6 subtraction of vectors
Using the parallelogram law of addition, a-b = a+(-b)
Example 3.4:

12. 3.7 scalar multiplication of vectors
Let k be a scalar and a represent a vector, the scalar multiplication ka is:
A vector whose length is |k| time that of the length a
Example 3.5:

14. 3.8 parallel vectors
Two nonzero vectors a and b are parallel (a||b) , if one is a scalar multiple of the other. If a= αb with α>0 , then a and b have the same direction. If α<0, then a and b have opposite direction.
E.g:

15. 3.9 component of vectors (unit vectors)
If we place the initial point of the vector u at the origin then the terminal point of u has coordinate <x, y, z>. These coordinate is called the components of u and we write
u = xi + yj + zk
where i, j and k are unit vectors in the x, y and z -axes, respectively:
Unit vector (i, j, k )
i = <1,0,0> be a unit vector in the OX direction (x-axis)
j = <0,1,0> be a unit vector in the OY direction (y-axis)
k = <0,0,1> be a unit vector in the OZ direction (z-axis)

16. Vector in 2D (R2)
Example 3.6:

17. 3.10 POSITION vectors Vector in 3D (R3) Example 3.7:
The position vector of the point with coordinates P(x, y, z) is
where r represent the vector from the origin to point P in R3 .

18. 3.11 component of vectors
If is a vector with initial point P(a, b, c) and terminal point Q(x, y, z), then
Example 3.8:
Exercise 3.1:

19. 3.12 magnitude (length) of vectors
The magnitude or length of any vector a =<x, y, z> is
Example 3.9:
Exercise 3.2:

20. 3.12 magnitude (length) of vectors
The magnitude or length of any vector from point P(a, b, c) to point Q(x, y, z) is defined as
Example 3.10:
Exercise 3.3:

21. 3.13 unit vectors
If u is a vector, then the unit vector in the direction of u is defined as
Example 3.11:
Exercise 3.4:

22. 3.15 operations of vectors by components
If a=<a1, a2, a3>, b=<b1, b2, b3>, and k is a scalar, then
Example 3.13:
Exercise 3.6:

23. Properties of a vector
If a, b and c are vectors, k and t are scalars,

24. 3.16 product of two vectors
The scalar/dot product of two vectors a=<a1, a2, a3> and
b=<b1, b2, b3>, denoted a . b, is
Example 3.14:
Exercise 3.7:

25. If θ is the angle between the vectors a and b , then
Theorem 1:
If θ is the angle between the vectors a and b , then
Note: a.b are orthogonal (perpendicular) if and only if a.b=0
Example 3.15:
Exercise 3.8:

26. Theorem 1: (Angle between two vectors)
If θ is the angle between the non-null vectors a and b , then
θ is an acute angle if and only if a.b>0
θ is an obtuse angle if and only if a.b<0
θ =90O is a right angle if and only if a.b=0
Example 3.15:

27. Properties of the Scalar/Dot Product:
If a and b are vectors, k is scalars, then

28. Exercise 1:

29. Exercise 2:

30. The cross product of two vectors a=<a1, a2, a3> and
b=<b1, b2, b3>, denoted a x b by the determinant of the matrix
Note: The result of vector product is a vector
Example 3.16:
Exercise 3.9:

31. If θ is the angle between the vectors a and b , then
Theorem 2:
If θ is the angle between the vectors a and b , then
Note: axb=0 if and only if a//b (a and b are parallel)
Example 3.17:

32. Exercise 3:

33. Properties of the Cross Product:
If a, b and c are vectors, k is scalars, then

34. 3.17 APPLICATIONS OF VECTORS
Projections
Equation of Planes
Parametric Equations of Line
Problems Involving Planes

35. Projections Consider the diagram below: Letθ Q S a
The length of the shadow
The scalar projection of vector b onto a (also called the component of b along a) is the length/magnitude of the straight line PS or which is

36. Consider the diagram below: LetP R b θ Q S a
Shadow of b onto a
The vector projection of vector b onto a is a vector in the direction of a with b magnitude equal to the length of the straight line PS or

37. Example 3.17:
Exercise 3.10:

38. Equations of Planes
Calculate the equation of the planes are defined as follows:
i) with normal vector, through the point (1,0,-1).
ii) through the origin and parallel to and
iii) through the three points (1,2,0), (3,1,1) and (2,0,0).
iv) parallel to the plane in (a) and passing through the point
(1,2,3).

39. Equations of Planes i) with normal vector, through the point (1,0,-1).
P(x0,y0,z0)
Q(x,y,z) n
Suppose that is plane with the vector and a normal vector to the plane n.
Let point P(x0,y0,z0) be an arbitrary point on and Q(x,y,z) be any point on plane .
normal vector, n is orthogonal

40. i) with normal vector, through the point (1,0,-1).
Q(x,y,z)
Then,

41. Exercise 4:

42. Example 3.20:
Exercise 3.13:

43. Equations of Planes ii) through the origin and parallel to and . n u v

44. Equations of Planes
iii) through the three points A(1,2,0), B(3,1,1) and C(2,0,0).
A(1,2,0)
B(3,1,1)
C(2,0,0)
Since A, B and C are coplanar, then the vector and are also coplanar. Thus will give a normal vector to the plane.

45. Equations of Planes
iii) through the three points A(1,2,0), B(3,1,1) and C(2,0,0).
A(1,2,0)
B(3,1,1)
C(2,0,0)

46. iv) parallel to the plane in (i) and passing through the point
(1,2,3).
Parallel to the plane from (i), x+2y+3z+2=0 and pass through the point (1,2,3)

47. Exercise 5:

48. Intersection of the plane
Given two intersecting planes with angle θ between them.
Let n1 and n2 be normal vector to these planes. Then,
Thus two planes are
Perpendicular if
Parallel if where k is scalar.

49. Intersection of the plane

50. Exercise 6:

51. Parametric Equations of a Line
Suppose L is a straight line that passes through the point P(x0,y0,z0) and is parallel to the vector as diagram below:
O(0,0,0) v
P(x0,y0,z0)
Q(x,y,z) x y z O P Q
Then another point Q(x,y,z) lies on the line L if and only if the vector v and are parallel:

52. Parametric Equations of a Line
Q(x,y,z)
P(x0,y0,z0)

53. Example 3.21:
Exercise 3.14:

54. Exercise 7:

55. Intersection of two lines
“intersecting”
“parallel”
“skew”=that the lines are neither
parallel nor intersecting

56. Example:
Exercise 7:

57. Problems involving planes
We shall discuss about the shortest distance from a point to a plane. We consider 3 cases as follows:
i) The distance from a point to a plane.
ii) The distance between two parallel planes..
iii) The distance between two skewed lined on the plane.
P0(x0,y0,z0) n D
P1(x,y,z) P D P D
(i)
(ii)
(iii)

58. i) The distance from a point to a plane.
P0(x0,y0,z0) D
P1(x,y,z)
projection of b onto normal vector, n.
Thus,
Since P0(x0,y0,z0) lies on the plane, then this point satisfy the equation of plane, that is
Thus, the formula D can be written as,
Let P0(x0,y0,z0) be a point on the plane and b be the vector corresponding to
Then,
From the figure above, the shortest distance, D from P1 to the plane is equal to the absolute value of scalar

59. ii) The distance between two planes
The shortest distance between two parallel planes
ax+by+cz+d1=0
and ax+by+cz+d2=0
is given by

60. Example 3.22:
Exercise 3.15:

61. ii) The distance between the skewed lines
Example:
Find the distance between the skewed lines as below, P1 D P2

62. Exercise 8: