1. GROUPS MEMBERSName Matric No.Ridwan bin shamsudin D20101037472Mohd. Hafiz bin Salleh D20101037433Muhammad Shamim BinD20101037460ZulkefliJasman bin Ronie D20101037474Hairieyl Azieyman Bin Azmi D20101037426Mustaqim Bin MusaD20101037402

2. y(3,2) (4,2) x Last but not least 3. WHAT IS VECTOR? VECTOR REPRESENTATIVE MAGNITUDE OF VECTOR NEGATIVE VECTOR ZERO VECTOR EQUALITY OF VECTOR PARALLEL VECTOR VECTOR MULTIPLICATION BY SCALAR NEXT 4. VECTOR ADDITION VECTOR SUBTRACTION DOT PRODUCTANGLE BETWEEN TWOVECTOR 5. WHAT IS . . 6. INTRODUCTION . . Vectoris a variable quantity that can be resolved into components. Vector also is a straight line segment whose length is magnitude and whose orientation in space is direction. Hurmm . . . 7. SCALAR VECTOR VECTOR PRODUCT A scalar quantityhas magnitude only A vector quantity haswith an appropriateboth magnitude andunit ofdirection.measurement. Examples of vector Example of scalarquantities are quantities arelength, speed, time, displacement,temperatue, mass velocity, accelerationand power. and force. 8. The most commonly used example of vectors in everyday life is velocity. Vectors also mainly used in physics and engineering to represent directed quantities. Vectors play an important role in physics about of a moving object and forces acting on it are all described by vectors. 9. Nice isnt it?? 10. Sinceseveral important physicalquantities are vectors, it is useful toagree on a way for representing themand adding them together. In the example involving displacement,we used a scale diagram in whichdisplacements were represented byarrows which were proportionatelyscaled and orientated correctly withrespect to our axes (i.e., the points of thecompass). 11. Thisrepresentation can be used for all vector quantities provided the following rules are followed: 1.The reference direction is indicated. 2.The scale is indicated. 3.The vectors are represented as arrows with alength proportional to their magnitude and are correctly orientated with respect to the reference direction. 4.The direction of the vector is indicated by an arrowhead. 5.The arrows should be labelled to show which vectors they represent. 12. For example, the diagram belowshows two vectors A and B, where Ahas a magnitude of 3 units in adirection parallel to the referencedirection and B has a magnitude of 2units and a direction 60 clockwise tothe reference direction:I see ~ 13. Thelength of a vector is called the magnitude or modulus of the vector. A vector whose modulus is unity is called a unit vector which has magnitude. The unit vector in the direction is called The unit vectors parallel to 14. The magnitude of vector a is written as |a|. The magnitude of vector AB is written as|AB|. 2 + 2 If a = then the magnitude |a|=*using pythagorean theorem. 15. EXAMPLES :1. Find the magnitude of the vector 2Solution : 16. A vector having the same magnitude but opposite direction to a vector A, is -A.If v is a vector, then -v is a vector pointing in the opposite direction.If v is represented by (a, b, c)T then -v is represented by (-a, -b, -c)T. 17. Example : Write down the negetive ofsolution : 3 = 2 3 = 2 18. Is a vector with zero magnitude andno direction|0|= 0 19. EXAMPLE : Determine whether w-y-x+z is a zerovector.SolutionFrom the diagram,w-y-x+z = OSince it does not has magnitude,thus it is a zerovector 20. 2 vectors u and v are equal if theircorresponding components are equal For example, if u=ai +bj and v=ci + dj then u = va=c and b=d Or in another word we can say it isequal if the vectors have samemagnitude and same direction 21. Example*note that =2i+j , =-2i-j 22. Vectors are parallel if they have thesame direction Both components of one vector mustbe in the same ratio to thecorresponding components of theparallel vector.(i) v1 kv2 , k any scalar (ii) v1 .v2 v1 v2 or v1 .v2 v1 v2 v x v 0 (iii) 12 23. EXERCISEExerciseGiven 2i-3j and 8i+yj are parallel vector. Find the value of y.SolutionSince they are parallel vectorsLet 8i+yj=k(2i-3j),k is any scalar 8i+yj=2ki-3kj8=2k y=-3kk=4 =-3(4) =-12 24. VECTOR MULTIPLICATION BYSCALAR 25. The scalar product(dot product) of twovectorsand is denoted byand defined as a b a b cosWhere is the angle between andwhich converge to a point or diverge from apoint. 26. m1m2 is an abtuse angle 27. Use this:a . a = a 2 28. Rule 1 29. Rule 2 30. Rule 3 31. SPECIAL CASE 32. Algebraic properties of thescalar product for any vectora, b and c and m is aconstant 33. 1) a . a = a 22) a . b = b . a3) a . (b + c) = a . b + a . c4) (a b )c) (a b c) a b c 5) m (a . b) = (ma) . b = (a . b)m 6) a . b = a b if and only if a parallel to ba . b = a b if and only if a and b inopposite direction7) a . b = 0 if and only if a is perpendicular to b8). 34. Example: Evaluate a) (2 i j ) (3 i 4 k )~ ~ ~ ~ b) (3 i 2 k ) (i 2 j 7 k )~ ~ ~ ~ ~ 35. SOLUTION EXAMPLE 1a) ~ ~ ~ ~ 2 i j 3i 4 k 23 10 04 6 36. b) 3 j 2 k i 2 j 7 k ~ ~ ~ ~ ~ 01 32 2 7 20 37. Definition of VectorMultiplication In Vector Multiplication, a vector ismultiplied by one or more vectors orby a scalar quantity. 38. More about VectorMultiplication There are three different types ofmultiplication: dot product, cross product,and multiplication of vector by a scalar. The dot product of two vectors u and v isgiven as u v = uv cos where is theangle between the vectors u and v. The cross product of two vectors u and vis given as u v = uv sin where isthe angle between the vectors u and v. When a vector is multiplied by a scalar,only the magnitude of the vector ischanged, but the direction remains thesame. 39. Examples of VectorMultiplication If the vector is multiplied by a scalar then=. If u = 2i + 6j and v = 3i - 4j are twovectors and angle between them is 60,then to find the dot product of thevectors, we first find their magnitude.Magnitude of vectorMagnitude of vectorThe dot product of the vectors u, v is u v = uv cos = (2 ) (5) cos 60= (2 ) (5) =5 40. If u = 5i + 12j and v = 3i + 6j are twovectors and angle between them is 60,then to find the cross product of thevectors, we first find their magnitude.Magnitude of vectorMagnitude of vectorThe cross product of the vectors u, v is u v = uv sin = (3 ) (13) sin 60= 39 (2)= 78 41. Solved Example on Vector Multiplication Which of the following is the dot product of thevectors u = 6i + 8j and v = 7i - 9j?Choices:A. 114B. - 30C. - 2D. 110Correct Answer: BSolution:Step 1: u = 6i + 8j, v = 7i - 9j are the two vectors.Step 2: Dot product of the two vectors u, v = u v= u1v1 + u2v2Step 3: = (6i + 8j) (7i - 9j)Step 4: = (6) (7) + (8) (- 9) [Use the definition ofthe dot product of two vectors.]Step 5: = - 30 [Simplify.] 42. Definition of Addition ofVectors Adding two or more vectors to form asingle resultant vector is known asAddition of Vectors. 43. More about Addition ofVectors If two vectors have the same direction,then the sum of these two vectors isequal to the sum of their magnitudes,in the same direction. If the two vectors are in oppositedirections, then the resultant of thevectors is the difference of themagnitude of the two vectors and is inthe direction of the greater vector. 44. Examples of Addition ofVectors To find the sum of the vectors of and , they are placed tail to tail to form two adjacent sides of a parallelogram and the diagonal gives the sum of the vectors and . This is also called as parallelogram rule of vector addition. 45. If the vector is represented inCartesian coordinate, then the sum ofthe vectors is found by adding thevector components.The sum of the vectors u = and v = is u + v == 46. Definition Of Subtraction OfVectors subtracting two or more vectors toform a single resultant vector is knownas subtraction of vectors. 47. example f the vector is represented inCartesian coordinate, then thesubtraction of the vectors is found bysubtracting the vector components.The sum of the vectors u = and v = is u - v == 48. The angle between 2 lines The two lines have the equations r = a+ tb and r = c + sd.The angle between the lines is foundby working out the dot product of band d. We have b.d = |b||d| cos A. 49. Example Find the acute angle between the linesL : r i 2 j t (2i j 2k )1L : r 2i j k s(3i 6 j 2k )2Direction Vector of L1, b1 = 2i j + 2kDirection Vector of L2, b2 = 3i -6j + 2kIf is the angle between the lines,(2i j 2k ).( 3i 6 j 2k )Cos =2i j 2k 3i 6 j 2k 50. EXAMPLE664Cos = 9 4916Cos =21 = 40 22