1. Engineering Fundamentals
Session 7 (3 hours)

2. Unit Vector A vector of length 1 unit is called a unit vector
i represents a unit vector in the direction of positive x-axis
j represents a unit vector in the direction of positive y-axis

3. Unit Vector Examples y ^ y -3i ^ ^ -2j i ^ ^ 2i+j ^ ^ ^ 4j 2i j ^ 5i x

4. A Vector in terms of i and j^ ^
A 2D vector can be written as r=ai+bj
modulus or magnitude (length or strength) of vector ^ ^ y x r a b i j

5. Addition of Vectors If
Then
E.g. y x b by ay
a+b a ax by

6. Subtraction of Vectors
Similarly, for
Then
E.g. y x ? by b ay a ax by y x a -b
a+(-b)

7. Exercise A = 2i + 3j, B= -i –j (bolded symbol denotes vectors)
A+B=______________
A-B=_______________
3A=_________________
|A| = ______________
the modulus of B______

8. Example ^ ^ ^ ^
If a=7i+2j and b=6i-5j, find a+b, a-b and modulus of a+b (bolded symbols denotes vectors)
Solution

9. Example
Find the x and y components of the resultant forces acting on the particle in the diagram
Solution: (Hint: the phase angles of the vectors are -15 and 210 degrees.) y x

10. Scalar Product of Vectors
Scalar product, or dot product, of 2 vectors: 
How does the dot product behave when a and b are perpendicular to one another ?
When a and b have the same direction?
Angle between the 2 vectors

11. Exercise i.i = _________ i.j=___________ j.j=__________
a.b = ___________ a 2
40 degrees
20 degrees 1 b

12. Scalar Product of Rectangular Vectors
For x-y coordinates,
It can be shown that

13. Exercise [3,5].[2,-1]=________
The dot product of –i + j and 2i-3j is ________________
The scalar product of 5i and 2i + j is _____________

14. Example If and Find , and angle between two vectors Solution:
Notice that a.b = b.a

15. Example (cont’d)

16. Scalar Product of 3D Rectangular Vectorsx y z
Similarly, for x-y-z coordinates,
Then az ax ay a bz bx by b

17. Exercise
Scalar product of 3i + 2j –k and –i + j = _______________

18. Scalar Product Properties
Properties of scalar product
Commutative:
Distributive:
For two vectors and , and a scalar k,

19. Exercise A = [1,2], B=[2,-3], C=[-4,5] A.(B+C) = _________
A.B + A.C = _________
3 A.B = __________
A. (3B) = ___________
(bolded symbols denotes vectors)

20. Scalar Product of Vectors
If two vectors are perpendicular to each other, then their scalar product is equal to zero.
i.e. if then
E.g. Given and
Show that and are mutually perpendicular
Solution:

21. Vector Product of Vectors
Vector product, or cross product, denoted
Defined as
The vector product of two vectors and is a vector of modulus in the direction of where is a unit vector perpendicular to the plane containing and in a sense (forward/backward direction) defined by the right-handed screw rule 

22. Right-Hand-Rule for Cross Product
a X b b a

23. Vector Product of Vectors
Note that
if Ө=0o, then
if Ө=90o, then
It can be proven that

24. Vector Product of Vectors
Properties of vector product
NOT commutative:
Distributive: 3.
Easy way to memorize #3: use right-hand rule

25. Example
Simplify
Solution:

26. Vector Product of Rectangular VectorsIf
then
E.g. Evaluate if and
Hence calculate

27. Example
Solution:
We know
Substitute

28. Concept Map in terms of matrix A = A/|A| [Vx, Vy] Rectangular form
unit vector
Vectors
in terms of i and j
i = unit vector in x direction
Vx i + Vy j
2D, 3D
magnitude=1
j = unit vector in y direction
cross product v vector X vector
Vector operations
k = unit vector in z direction
results in vector
AyBz-AzBy i
-(AxBz-AzBx) j
+(AxBy-AyBx) k
vector + vector , vector - vector, scalar X vector
Dot product vector.vector
Direction: right-hand rule
results in scalar
A.B=|A| |B| cos θ
|AXB|
= |A| |B| sin θ
A.B = Ax Bx + Ay By
A.B = Ax Bx + Ay By + Az Bz