The scalar and Vector presentation for 11th class students
- 1. © T Madas 1 2 AB BC CD ( ) + + = + + + a b a b u u r u u r u u r 2 3 + i j
- 2. © T Madas A vector is a line with a start and a finish. It therefore has: 1. line of action 2. a direction 3. a given size (magnitude) A B AB u u u r = a A B BA u u r - = a
- 3. © T Madas a AB u u u r a we write vectors in the following ways: By writing the starting point and the finishing point in capitals with an arrow over them With a lower case letter which: is printed in bold or underlined when handwritten In component form if the vector is drawn on a grid: 4 5
- 4. © T Madas E F A B C D G H Let AB = a u u u r CD = u u u r 2a EF = u u r 1 2 a HG = u u u r - 2a
- 5. © T Madas 4 5 A B -5 4 C D AB = CD =
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- 7. © T Madas A B C D Let AB = a u u u r Let AD = b u u u r a a b b ABCD is a parallelogram DC = u u u r a BC = u u u r b
- 8. © T Madas A B C D Let AB = a u u u r Let AD = b u u u r a a b b ABCD is a parallelogram DC = u u u r a BC = u u u r b Now adding vectors AC = u u u r AD u u u r DC + = u u u r + a b = + a b AB = u u u r BC + = u u u r + b a
- 9. © T Madas A B C D Let AB = a u u u r Let AD = b u u u r a a b b ABCD is a parallelogram DC = u u u r a BC = u u u r b Now adding vectors BD = u u u r BA = u u r AD + = u u u r + b - a = - b a BC u u u r CD + = u u u r - a b
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- 11. © T Madas A B C M ABC is a triangle N with M the midpoint of AB and N the midpoint of BC.Let AB = a u u u r and AC = b u u u r AM = u u u r 1 2 a MB = u u u r BC = u u u r BA u u r AC + = u u u r + b - a = - b a 1 2 a 1 2 a 1 2 a b
- 12. © T Madas A B C M ABC is a triangle N with M the midpoint of AB and N the midpoint of BC.Let AB = a u u u r and AC = b u u u r AM = u u u r 1 2 a MB = u u u r BN = u u u r NC = u u u r BC = u u u r BA u u r AC + = u u u r + b - a = - b a 1 2 a 1 1 2 2 - b a 1 1 2 2 - b a 1 2 a 1 2 a 1 1 2 2 - b a 1 1 2 2 - b a b
- 13. © T Madas A B C M ABC is a triangle N with M the midpoint of AB and N the midpoint of BC.Let AB = a u u u r and AC = b u u u r AM = u u u r 1 2 a MB = u u u r BN = u u u r NC = u u u r MN = u u u r MB u u u r BN + = u u u r 1 1 2 2 - + b a 1 2 a 1 2 a 1 1 2 2 - b a 1 1 2 2 - b a 1 2 a 1 1 2 2 - b a 1 1 2 2 - b a 1 2 = b 1 2 b b 1 2 a What is the relationship between AC and MN ?
- 14. © T Madas 1 2 - b A B C M ABC is a triangle N with M the midpoint of AB and N the midpoint of BC.Let AB = a u u u r and AC = b u u u r AM = u u u r 1 2 a MB = u u u r BN = u u u r NC = u u u r NA = u u r NM u u u r MA + = u u u r 1 2 - b 1 2 - a 1 2 a 1 1 2 2 - b a 1 1 2 2 - b a 1 2 a 1 1 2 2 - b a 1 1 2 2 - b a 1 2 - ( ) = 1 2 b b 1 2 a b + a 1 2 - ( ) = + a b NA = u u r NB u u u r BA + = u u r - a 1 2 + a 1 2 - = b 1 2 - a 1 2 b NA = u u r NC u u u r CA + = u u r - b 1 2 - a 1 2 - = b 1 2 - a etc etc etc etc
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- 16. © T Madas ABCDEF is a regular hexagon. , and . AB BC CD = = = a b c u u u r u u u r u u u r A B C D M is the midpoint of CE. E F M a b c Write and simplify expressions in terms of a, b and c for : (a) , CE u u u r (b) MD u u u r and (c) MB u u u r a b c CE = u u u r CD u u u r DE + = u u u r c- a CM = u u u r ME = u u u r 1 2 c 1 2 - a 1 1 2 2 - c a MD = u u u r ME u u u r ED + u u u r 1 2 = c 1 2 - a + a 1 2 = c 1 2 + a solution
- 17. © T Madas ABCDEF is a regular hexagon. , and . AB BC CD = = = a b c u u u r u u u r u u u r A B C D M is the midpoint of CE. E F M a b c Write and simplify expressions in terms of a, b and c for : (a) , CE u u u r (b) MD u u u r and (c) MB u u u r a b c 1 1 2 2 - c a MB = u u u r MC u u u r CB + u u r 1 2 - = c 1 2 + a - b 1 2 - c 1 2 = a - b 1 2 ( ) = a - c 2 - b 1 2 ( 2 ) = - - a b c solution
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- 19. © T Madas M A B C D N ABCD is a parallelogram. and . AB AD = = a b u u u r u u u r M is the midpoint of AD Write and simplify expressions in terms of a and b for : (a) , BD u u u r (b) BN u u u r and (d) NC u u u r N is a point of BD so that BN : ND = 2 : 1. (c) MN u u u r a 1 2 b 1 2 b a b 2 2 3 3 - b a solution BD = u u u r BA u u r AD + = u u u r - a+ b = - b a BN = u u u r 2 3 BD = u u u r 2 3 - a 2 3 b
- 20. © T Madas M A B C D N ABCD is a parallelogram. and . AB AD = = a b u u u r u u u r M is the midpoint of AD Write and simplify expressions in terms of a and b for : (a) , BD u u u r (b) BN u u u r and (d) NC u u u r N is a point of BD so that BN : ND = 2 : 1. (c) MN u u u r a 1 2 b 1 2 b a b MN = u u u r MA u u u r BN + = u u u r 1 2 - b + a 2 2 3 3 - b a AB + u u u r 2 2 3 3 é ù + - ê ú ë û b a 1 3 = a 1 6 + b solution
- 21. © T Madas M A B C D N ABCD is a parallelogram. and . AB AD = = a b u u u r u u u r M is the midpoint of AD Write and simplify expressions in terms of a and b for : (a) , BD u u u r (b) BN u u u r and (d) NC u u u r N is a point of BD so that BN : ND = 2 : 1. (c) MN u u u r a 1 2 b 1 2 b a b NC = u u u r NB u u u r 2 3 - b 2 3 + a 2 2 3 3 - b a BC + = u u u r + b 2 3 = a 1 3 + b (e) Using your answers from parts (c) and (d), show that M, N and C lie on a straight line solution
- 22. © T Madas M A B C D N ABCD is a parallelogram. and . AB AD = = a b u u u r u u u r M is the midpoint of AD Write and simplify expressions in terms of a and b for : (a) , BD u u u r (b) BN u u u r and (d) NC u u u r N is a point of BD so that BN : ND = 2 : 1. (c) MN u u u r a 1 2 b 1 2 b a b 2 1 3 3 NC = + a b u u u r 2 2 3 3 - b a (e) Using your answers from parts (c) and (d), show that M, N and C lie on a straight line 1 1 3 6 MN = + a b u u u r 2 1 3 3 NC = + a b u u u r 1 3 ( ) = 2a + b 1 1 3 6 MN = + a b u u u r 1 6 ( ) = 2a + b What is the ratio MN : NC ? solution
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- 24. © T Madas 6 and 6 . AB AD = = a b u u u r u u u r ABCD is a parallelogram.M is the midpoint of AB a) Find the vector in terms of a and b b) Prove that MNC is a straight line 1 3 BN BD = u u u r u u u r N is a point of BD so that . A B C D M NC u u u r 3a 3a 6b 6a 6b BD = u u u r BC u u u r BD + = u u u r 6b 6 - a BN = u u u r 1 3 BD = u u u r 2b 2 - a 2 2 - b a N
- 25. © T Madas 1 3 BN BD = u u u r u u u r N is a point of BD so that . 6 and 6 . AB AD = = a b u u u r u u u r ABCD is a parallelogram.M is the midpoint of AB a) Find the vector in terms of a and b b) Prove that MNC is a straight line A B C D M NC u u u r 3a 3a 6b 6a 6b 2 2 - b a NC = u u u r NB u u u r - (2 2 ) - b a BC + = u u u r 6 + b - 2 = b 2 + a 6 + b 2 = a 4 + b 2 4 + a b N
- 26. © T Madas 2 + a b 1 3 BN BD = u u u r u u u r N is a point of BD so that . 6 and 6 . AB AD = = a b u u u r u u u r ABCD is a parallelogram.M is the midpoint of AB a) Find the vector in terms of a and b b) Prove that MNC is a straight line A B C D M N NC u u u r 3a 3a 6b 6a 6b 2 2 - b a 2 4 + a b MN = u u u r MB u u u r (2 2 ) + - b a BN + = u u u r 3a 2 - a 3 = a 2 + b = a 2 + b
- 27. © T Madas 2 + a b 1 3 BN BD = u u u r u u u r N is a point of BD so that . 6 and 6 . AB AD = = a b u u u r u u u r ABCD is a parallelogram.M is the midpoint of AB a) Find the vector in terms of a and b b) Prove that MNC is a straight line A B C D M N NC u u u r 3a 3a 6b 6a 6b 2 2 - b a 2 4 + a b 2 MN = + a b u u u r 2 4 NC = + a b u u u r MC = u u u r 3 6 + a b , and have the same direction MN NC MC u u u r u u u r u u u r , and are collinear M N C
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- 29. © T Madas ABCD is a quadrilateral and M, N , P and Q are the midpoints of AB, BC, CD and DA respectively. AM = a, BN = b and CP = c. Find in terms of a, b and c: a) AD b) AQ c) MQ d) NP e) Deduce a geometric fact about the quadrilateral MNPQ A B C D M N P Q a b c a b c AD = AB + BC + CD = 2a + 2b + 2c = 2(a + b + c) AQ = a + b + c a+b+c a+b+c
- 30. © T Madas ABCD is a quadrilateral and M, N , P and Q are the midpoints of AB, BC, CD and DA respectively. AM = a, BN = b and CP = c. Find in terms of a, b and c: a) AD b) AQ c) MQ d) NP e) Deduce a geometric fact about the quadrilateral MNPQ A B C D M N P Q a b c a b c MQ = MA + AQ = -a + (a + b + c) = b + c a+b+c a+b+c b+c NP = NC + CP = b + c b+c
- 31. © T Madas ABCD is a quadrilateral and M, N , P and Q are the midpoints of AB, BC, CD and DA respectively. AM = a, BN = b and CP = c. Find in terms of a, b and c: a) AD b) AQ c) MQ d) NP e) Deduce a geometric fact about the quadrilateral MNPQ A B C D M N P Q a b c a b c a+b+c a+b+c b+c b+c A quadrilateral with a pair of sides equal and parallel is a parallelogram or can show that MN = QP = a + b Hence MNPQ is a parallelogram.
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