Scalar and vector quantities

GROUP PRESENTATION

1. SCALAR AND VECTOR QUANTITIES RAFIA (LEADER) MUHAMMAD FARAZ SYED WAMIQ HUSSAIN ISMAIL UZAIR MATEEN Prepared by: MUHAMMAD FARAZ

2. PHYSICAL QUANTITIES The observable and measurable quantities are called physical quantities. TYPES OF PHYSICAL QUANTITIES: Scalar quantities Vector quantities

3. SCALAR QUANTITIES Those quantities which have magnitude without directions. EXAMPLES: Mass Speed work Energy temperature

4. VECTOR QUANTITIES Physical quantities those are represented by a number (magnitude) with appropriate unit and a particular direction are called as vector . Examples: Displacement. Force. Momentum. Torque.

5. EQUAL VECTOR: Such vector, whose magnitude and direction is same as that of magnitude and directions of a given position vector, is called equal to the position vector”.

6. ZERO VECTOR Such vectors whose magnitude is zero and which has no direction or which may have all directions is called null vector.

7. ADDITION OF VECTOR A process in which two or more vectors are combine to get a resultant vector is called addition of vector.

8. SUBTRACTION OF VECTOR Similarly orientated vectors can be subtracted the same manner.

9. METHOD OF VECTOR ADDITION GRAPHICAL METHOD ANALYTICAL METHOD GRAPHICAL METHOD: HEAD TO TAIL RULE PARALLELOGRAM METHOD

10. HEAD TO TAIL RULE Two vectors A and B are added by drawing the arrows which represent the vectors in such a way that the initial point of B is on the terminal point of A . The resultant R=A+B+C , is the vector from the initial point of A to the terminal point of C . R = A +B+C

11. PARALLELOGRAM METHOD In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin and the parallelogram constructed as follows:

12. The resultant R is the diagonal of the parallelogram drawn from the common origin.

13. RESOLUTION OF VECTOR The process of splitting a vector into various parts or components is called "RESOLUTION OF VECTOR" These parts of a vector may act in different directions and are called "components of vector". We can resolve a vector into a number of components .Generally there are three components of vector viz. Component along X-axis called x-component Component along Y-axis called Y-component Component along Z-axis called Z-component Here we will discuss only two components x-component & Y-component which are perpendicular to each other.These components are called rectangular components of vector.

14. ANALYTICAL METHOD RESOLVING VECTOR INTO RECTANGULAR COMPONENTS: Consider a vector acting at a point making an angle q with positive X-axis. Vector is represented by a line OA. From point A draw a perpendicular AB on X- axis.Suppose OB and BA represents two vectors. Vector OA is parallel to X-axis and vector BA is parallel to Yaxis.Magnitude of these vectors are V x and V y respectively.By the method of head to tail we notice that the sum of these vectors is equal to vector .Thus V x and V y are the rectangular components of vector . V x = Horizontal component of . V y = Vertical component of .

15. RESOLVING VECTOR INTO RECTANGULAR COMPONENTS MAGNITUDE OF HORIZONTAL COMPONENT Consider right angled triangle DOAB Consider right angled triangle DOAB MAGNITUDE OF VERTICAL COMPONENT

16. RESOLVING VECTOR INTO RECTANGULAR COMPONENTS MAGNITUDE OF VECTOR: The magnitude of A, |A| , can be calculated from the components, using the Theorem of Pythagoras: DIRECTION OF VECTOR: The direction can be calculated using trignometric ratio.

17. DOT OR SCALOR PRODUCT If the product of two vector is a scalar quantity is called scalar or dot product. The dot product of two vector making angle b/w them is given by; A . B = ABCosO A .B =A( BCosO ) EXAMPLE: W=FS OR P=FV The product of two vector force and displacement is work which is scalar quantity. PROPERTIES: A . B=B . A A . B = O if O = 90 A . B = AB(max) if O =O i . i = j . j = k . k = 1 i . j = j . k = k . i = O

18. QUESTION Q: Find work If S = 3i + 2j – 5k and F =2i – j - k ANS: W = FS W = (2i – j – k ) . (3i + 2j -5k ) W =6 ( i . i ) -2 (j . J) +5 (k . K) W = 6 (1) -2 (1) + 5 (k . K) W = 6 – 2 + 5 W = 9 units

19. CROSS OR VECTOR PRODUCT If the product of two vector is a vector quantity product is called vector or cross product. MATHEMATICALLY, A B = AB SinO n Where ABSinO is magnitude of A B and n is a unit vector that give direction of A B and direction perpedicular to both A and B which can be find by R . H . RULE. PROPERTIES: A B = B A A B = AB (max) when O=90 A B = O if O =O i i = j j = k k = O i j = k , j k = i , k i = j

20. EXAMPLE: = F S Force and displacement is a vector quantity and its result TORQUE is also a vector. QUESTION Q: Find cross product of A & B if A =2i – 3j – k , B = i +4j – 2k ? ANS.