1. ME 2304: 3D Geometry & Vector Calculus Dr. Faraz Junejo

2. Assessments  3 Quizzes: 9 Marks (3 Marks each) You will be notified 1 Week in advance  2 Assignments: 11 Marks (5.5 Marks each)  Mid Term Exam (8 th Week – 19 th Sept): 30 Marks  Final Exam (17 th Week – 21 st Nov) : 50 Marks

3. Class Conduct  Absences from class  Late in class  Assignment submission  Absence from Mid Term or Final Exam  Cheating  Anything you want to ask, ASK NOW !!

4. A scalar quantity is a quantity that has magnitude only and has no direction in space Scalars Examples of Scalar Quantities:  Length, i.e. 5m  Area  Volume  Time  Mass  Speed  Temperature

5. A vector quantity is a quantity that has both magnitude and a direction in space Vectors Examples of Vector Quantities:  Displacement  Velocity  Acceleration  Force

6. More about Vectors A vector is represented on paper by an arrow 1. the length represents magnitude 2. the arrow faces the direction of motion 3. a vector can be “picked up” and moved on the paper as long as the length and direction its pointing does not change

7. Position Vector If A is the point (a 1,a 2 ), as shown in figure given below, then is called position vector for (a 1,a 2 ), or for point A. The numbers (a 1,a 2 ) are called components of vector. A(a 1,a 2 ) O Position vector x y a Terminal Point Initial Point

8. Position Vector (contd.) The magnitude of the vector a (a 1,a 2 ) is actually the length of its position vector is given by: Exercise: Sketch the position vectors for a=(-3,5), b=(0,-3) and find magnitude of each vector.

9. Vector diagrams are shown using an arrow The length of the arrow represents its magnitude The direction of the arrow shows its direction Vector Diagrams

10. Vectors in opposite directions: 6 m s -1 10 m s -1 =4 m s -1 6 N10 N=4 N Resultant of Two Vectors Vectors in the same direction: 6 N4 N=10 N 6 m =10 m 4 m TThe resultant is the sum or the combined effect of two vector quantities

11. Algebraic Vector Addition The sum of the two vectors, say a (a 1,a 2 ) and b(b 1,b 2 ) is obtained by simply adding the corresponding components, i.e. a+b=[a 1 +b 1, a 2 +b 2 ]; Basic properties of Vector addition are: 1.a + b = b + a 2.(a + b) + c = a + (b + c) 3.a + 0 = 0 + a = a 4.a + (-a)= 0, where –a represents a vector having the length and direction opposite to that of a

12. Exercise If a(4,-6) and b=(-5,8), find the vectors 2a+3b a-b 2a-3b

13. Exercise 1. Find The terminal point of the vector If its initial point is (-3,10) 2. Find The initial point of the vector If its terminal point is (4,7) Answer: (9, 8)

14. Important You can add vectors in any order and yield the same resultant.

15. The Parallelogram Law  When two vectors are joined tail to tail  Complete the parallelogram  The resultant is found by drawing the diagonal  When two vectors are joined head to tail  Draw the resultant vector by completing the triangle

16. Solution: Problem: Resultant of 2 Vectors CComplete the parallelogram (rectangle) θ TThe diagonal of the parallelogram ac represents the resultant force Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? 5 N 12 N 5 12 a bc d  The magnitude of the resultant is found using Pythagoras’ Theorem on the triangle abc RResultant displacement is 13 N 67 º with the 5 N force 13 N

17. 45º 5 N 90 º θ Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. Problem: Resultant of 3 Vectors 5 N 5 5 Solution: FFind the resultant of the two 5 N forces first (do right angles first) a b cd 7.07 N 10 N 135º NNow find the resultant of the 10 N and 7.07 N forces TThe 2 forces are in a straight line (45 º + 135 º = 180 º ) and in opposite directions SSo, Resultant = 10 N – 7.07 N = 2.93 N in the direction of the 10 N force 2.93 N

18. When resolving a vector into components we are doing the opposite to finding the resultant We usually resolve a vector into components that are perpendicular to each other Resolving a Vector Into Perpendicular Components y v x HHere a vector v is resolved into an x component and a y component

19. Here we see a table being pulled by a force of 50 N at a 30 º angle to the horizontal When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N Practical Applications 50 N y=25 N x=43.3 N 30 º  We can see that it would be more efficient to pull the table with a horizontal force of 50 N

20. If a vector of magnitude v and makes an angle θ with the horizontal then the magnitude of the components are: x = v Cos θ y = v Sin θ Calculating the Magnitude of the Perpendicular Components v y=v Sin θ x=v Cos θ θ y  Proof: x

21. Unit Vector A unit vector is a vector that has a magnitude of 1, with no units. Its only purpose is to point i.e. they are used to specify a direction We will use i, j for our unit vectors i means x – direction and j is y – direction We also put little “hats” (^) on i, j to show that they are unit vectors

22. Unit Vector (Cont)

23. If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value. If we want to find the unit vector having the same direction as w we need to divide w by 5. Let's check this to see if it really is 1 unit long. Unit Vector (cont)

24. If we know the magnitude and direction of the vector, let's see if we can express the vector in ai + bj form. As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction. Exercise

25. What is a scalar quantity? What is a vector quantity? How are vectors represented? What is the resultant of 2 vector quantities? What is the triangle law? What is the parallelogram law? What is unit vector ? Recap