This document discusses sinusoidal waves and vector functions. It defines key concepts for sinusoidal waves like amplitude, period, phase shift, and frequency. It also explains vector notation and properties, including adding vectors using the nose-to-tail and parallelogram methods. Vectors in three dimensions are represented using unit vectors i, j, and k. Vector operations like the scalar and vector products are also introduced.

1. Sinusoidal Waves
Learning Outcome 3

2. LO3: Use analytical and computational methods for solving problems
by relating sinusoidal wave and vector functions to their respective
engineering application.
Sinusoidal waves:
1. Sine waves and their applications
2. Trigonometric and hyperbolic identities
Vector functions:
1. Vector notation and properties
2. Representing quantities in vector form
3. Vectors in three dimensions

3. Sinusoidal Waveforms
y = Sin A

4. Amplitude, Period, Phase Shift and Frequency
• The Amplitude is the height from the center line to the peak. Or we
can measure the amplitude from highest to lowest points and divide
that by 2.
• The Period goes from one peak to the next (or from any point to the
next matching point)

5. ●The frequency of a sine wave is the number of complete cycles that happen
every second.
Find the frequency of below given wave form.
Exercise 01

6. ●The Phase Shift is how far the function is shifted horizontally from the usual position.
• The Vertical Shift is how far the function is shifted vertically from the usual position.

7. Sinusoidal Function
• Amplitude is A
• Period is 2π/B
• Phase shift is C (positive is to the left)
• Vertical shift is D
Definition
y = A sin (B(x+c))+D
Important Equations
• f = 1/T
• Ꙍ = 2πf

8. Define Amplitude, time period, frequency and phase shift.
1. y = sin(x)
2. y = 2 sin(4(x − 0.5)) + 3
3. y = 3 sin(100t + 1)
Exercise 02

9. Applications of Sinusoidal waveforms
●Natural vibrations
●Alternating current

10. Vector Functions
Scalar
Quantities such as time, temperature and mass are entirely defined by a numerical value and
are called scalars or scalar quantities.
Example:
the temperature in a room may be measured as, say, 16◦C, or the mass of a bearing may be
measured as, say, 3 kg.
Vector
Quantities such as velocity, force and acceleration, which have both a magnitude and a
direction, are called vectors.
Example:
The velocity of a car is 90 km/h due west, or a force of 20 N acts vertically downwards, or an
acceleration of 10m/s 2 acts at 50◦ to the horizontal.

11. Vector Representation
●A vector quantity can be represented graphically by a line, drawn so that:
a) The length of the line denotes the magnitude of the quantity.
b) The direction of the line denotes the direction in which the vector quantity acts.

12. Example:
1. A force of 9 N acting at 45° to the horizontal
2. A velocity of 20 m/s at −60° Note that an angle of −60° is drawn from the horizontal
and moves clockwise.
+45° Anticlockwise Direction
-60° Clockwise Direction

13. There are a number of ways of representing vector quantities. These include:
1) Using bold print
2) AB where an arrow above two capital letters denotes the sense of direction, where A is
the starting point and B the end point of the vector.
3) AB or a i.e. a line over the top of letters
4) a i.e. an underlined letter
AB AB
a a

14. Two Equal Vectors
If two vectors, a and b, are said to be equal, they have the same
magnitude and the same direction.
If a = b, then
a) a = b (magnitudes equal)
b) The direction of a = direction of b (The two vectors are parallel
and in the same direction as shown in figure.
Similarly, if two vectors a and b are such that b = -a, what can we say about their magnitudes?
their directions?
• Magnitudes are equal
• But the directions are opposite.

15. Addition of Vectors
Nose-to-tail method

16. There are two vectors called p and q.
P = a force of 40N, acting in the direction due East
q = a force of 30N, acting in the direction due North.
The sum of p and q vectors are called vector r.
Find the magnitude and direction of vector r.
Exercise 01

17. The Sum of number of vectors a + b + c + d +….

18. What is the sum of vectors a, b, c, d and the sum of vectors a, b, c, d, e??
ABCD is a quadrilateral, with G and H the mid-points of DA and BC respectively. Show
that AB + DC = 2GH
Exercise 01

19. Components of a Vector in terms of unit vector.
To add the two force vectors F1 and F2
Parallelogram Method
Steps:
1. Construct line BC parallel to OA
2. Construct line AC parallel to OB
3. The resultant force will be given by the
diagonal of the parallelogram i.e. OC
This procedure is called parallelogram method

20. A force of 5N is inclined at an angle of 45° to a second force of 8N, both forces acting at a
point. Find the magnitude of the resultant of these two forces and the direction of the
resultant with respect to the 8N force by:
1) The ‘Nose-to-tail’ Method
2) The ‘Parallelogram’ Method
Exercise 01

21. Examples
●

22. Vectors in Space
●A method of completely specifying the direction of a vector in space relative to some
reference point is to use three unit vectors, i, j and k, mutually at right angles to each
other.
Vector OP is defined by its components
a along OX
b along OY
c along OZ
I = unit vector in OX direction
j = unit vector in OY DIRECTION
K = unit vector in OZ direction

23. Direction Cosines
The direction of a vector in three dimensions is determined by the angles which the vector
makes with the three axes of references.

24. Scalar Product of Two Vectors (Dot Product)
●

25. Vector Product of Two vectors
●The vector product of a and b is written a x b and is denoted by,
●The product of vectors acts in a direction perpendicular to both a and b in such a
sense that a, b and a x b form a right-handed set in that order.
Note that b x a reverse the direction of rotation and the product vector would
now act downwards.
b x a = -(a x b )

26. ●