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  # INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 1
  
  TAJUK : VEKTOR 1
  
  NOTA RINGKAS
  
  Jika vektor AB a = maka vektor BA a =
  
  Hukum Segitiga
  
  AB AO OB = +
  AO AB OB AB BO = = +
  
  Vektor dalam Koordinat Cartesian
  Jika O(0 , 0) dan A(x , y), maka OA xi yj = +
  atau
  x
  OA
  y
  | |
  =
  |
  \ .
  dan AO xi yj = atau
  x
  AO
  y
  | |
  =
  |
  
  \ .
  .
  2 2
  OA x y = +
  
  Vektor unit OA adalah
  2 2
  xi yj
  OA
  OA x y
  +
  =
  +
  
  AKTIVITI
  
  Ungkapkan vektor dalam bentuk xi yj + dan
  x
  y
  | |
  |
  \ .
  .
  Example :
  
  Solution :
  
  1
  3
  3
  OA i j
  | |
  = + =
  |
  \ .
  
  1
  3
  3
  AO i j
  | |
  = =
  |
  
  \ .
  
  5
  5 5
  5
  OB i j
  | |
  = + =
  |
  \ .
  
  5
  5 5
  5
  BO i j
  | |
  = =
  |
  
  \ .
  
  ( 3 ) (5 5 )
  4 2
  AB AO OB
  i j i j
  i j
  = +
  = + +
  = +
  atau
  
  1 5
  3 5
  4
  2
  | | | |
  = +
  | |
  
  \ . \ .
  | |
  =
  |
  \ .
  
  0
  6
  4
  2
  2 4 6
  A(1,3)
  B(5,5)
  i
  
  j
  
  O A
  B
  A B
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 2
  
  Exercise :
  
  Solutions :
  
  OA
  
  =
  
  OB =
  
  OC =
  
  OD =
  
  AB =
  
  BC =
  
  BC =
  
  CB =
  BD =
  
  DB =
  CD =
  
  DC
  
  Find the magnitude for each of the vectors
  Example :
  
  2 2
  3 2
  (3) ( 2)
  AB i j
  AB
  =
  = +
  
  13 = unit
  
  (a) 4 3 PQ i j = +
  (b) 12 5 a i j =
  
  (c) 8 2 b i j = +
  (d) 5 3 OA i j = +
  
  (e) 5 12 AB i j =
  B(5,2)
  2
  0
  4
  -2
  -4
  -4 -2 2 4
  A(-5,2)
  C(-3,-2)
  D(4,-5)
  B(5,2)
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 3
  
  (f) 6 8 OB i j =
  
  (g) 3 8 BO i j =
  
  Find the unit vector for each of the following
  Example :
  
  2 2
  3 2
  3 2
  (3) ( 2)
  3 2
  13
  AB i j
  i j
  AB
  i j
  =
  
  =
  +
  
  =
  
  (a) 2 6 a i j =
  
  2 6
  40
  i j
  
  (b) 8 6 PQ i j = +
  
  8 6
  10
  i j +
  
  (c) 12 5 r i j = +
  
  12 5
  13
  i j +
  
  (d)
  3
  12
  a
  | |
  =
  |
  \ .
  
  3 1
  12
  153
  | |
  |
  \ .
  
  (e)
  8
  7
  b
  | |
  =
  |
  
  \ .
  
  8 1
  7
  113
  | |
  |
  
  \ .
  
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 4
  
  Find the following vector
  Example :
  
  2 3 a i j =
  
  5 b i j = +
  Solution :
  
  2 2(2 3 ) 4 6 a i j i j = =
  
  3 3( 5 ) 3 15 b i j i j = + = +
  
  2 2(2 3 ) ( 5 )
  4 6 5
  3
  a b i j i j
  i j i j
  i j
  + = + +
  = +
  =
  
  Exercise :
  
  2 4 a i j =
  
  3 5 b i j = +
  
  2a =
  
  3b =
  
  2 a b =
  
  3 2 b a =
  
  2a b =
  
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 5
  
  EXERCISE
  1. Rajah menunjukkan dua vektor OA dan OB dengan O adalah asalan.
  
  Ungkapkan
  (a) OB dalam bentuk
  x
  y
  | |
  |
  \ .
  ,
  (b) dalam bentuk xi yj + ,
  (i) OA
  (ii) AB
  
  [(a)
  5
  3
  | |
  |
  \ .
  (b) (i) 8 4 i j + (b) 13i j ]
  
  2. Rajah menunjukkan dua vektor OP dan OQ.
  
  Ungkapkan
  (a) OQ dalam bentuk xi yj + ,
  (b) PQ dalam bentuk
  x
  y
  | |
  |
  \ .
  .
  
  [(a) 8 8 i j (b)
  4
  10
  | |
  |
  
  \ .
  ]
  
  y
  Q(8, -8)
  P(4, 2)
  O
  x
  y
  A(8, 4)
  B(5, 3)
  O
  x
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 6
  
  3. Diberi vektor 5 12 AB i j = . Cari
  (a) AB ,
  (b) vektor unit dalam arah AB.
  
  [(a) 13 (b)
  5 12
  13
  i j
  ]
  
  4. Diberi dua vektor 4 OP i j = + dan 5 7 OQ i j = + .
  (a) Ungkapkan vektor PQ dalam bentuk
  x
  y
  | |
  |
  \ .
  .
  (b) Vektor unit dalam arah PQ .
  
  [(a) 6 3 i j + (b)
  6 3
  45
  i j +
  ]
  
  5. Diberi O(0 , 0), A(7 , -4) dan B(2 , 8). Cari
  (a) AB dalam bentuk
  x
  y
  | |
  |
  \ .
  ,
  (b) nilai AB .
  
  [(a)
  5
  12
  | |
  |
  \ .
  (b) 13]
  
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 7
  
  6. Diberi O(0 , 0), P(-4 , 8) dan Q(5 4). Cari
  (a) PQ dalam bentuk xi yj + ,
  (b) vektor unit dalam arah PQ .
  
  [(a) 9 12 i j (b)
  9 12
  15
  i j
  ]
  
  7. Diberi dua vektor 2 OA i j = dan 5 OB i j = + . Tentukan
  (a) AB ,
  (b) vektor unit dalam arah AB.
  
  [(a) 4 3 i j + (b)
  4 3
  5
  i j +
  ]
  
  8. Diberi bahawa 2 a i j = and 4 3 b i j = . Cari
  (a) vektor 2a b ,
  (b) vektor unit dalam arah 2a b .
  
  [(a) 2i j (b)
  2 7
  5
  i j
  ]
  
  ### Dapatkan aplikasi gratis kami
  
  Hak cipta © 2021 Scribd Inc.
explanation: The document is a mathematics intervention module for Additional Mathematics from the Johor Education Department, focusing on the topic of Vectors. It includes notes, examples, and exercises with solutions. The document contains multiple questions and answers related to vector operations, such as expressing vectors in different forms, finding magnitudes, and unit vectors.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
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distilabel_metadata.content: Anda di halaman 1dari 7
  
  # INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 1
  
  TAJUK : VEKTOR 1
  
  NOTA RINGKAS
  
  Jika vektor AB a = maka vektor BA a =
  
  Hukum Segitiga
  
  AB AO OB = +
  AO AB OB AB BO = = +
  
  Vektor dalam Koordinat Cartesian
  Jika O(0 , 0) dan A(x , y), maka OA xi yj = +
  atau
  x
  OA
  y
  | |
  =
  |
  \ .
  dan AO xi yj = atau
  x
  AO
  y
  | |
  =
  |
  
  \ .
  .
  2 2
  OA x y = +
  
  Vektor unit OA adalah
  2 2
  xi yj
  OA
  OA x y
  +
  =
  +
  
  AKTIVITI
  
  Ungkapkan vektor dalam bentuk xi yj + dan
  x
  y
  | |
  |
  \ .
  .
  Example :
  
  Solution :
  
  1
  3
  3
  OA i j
  | |
  = + =
  |
  \ .
  
  1
  3
  3
  AO i j
  | |
  = =
  |
  
  \ .
  
  5
  5 5
  5
  OB i j
  | |
  = + =
  |
  \ .
  
  5
  5 5
  5
  BO i j
  | |
  = =
  |
  
  \ .
  
  ( 3 ) (5 5 )
  4 2
  AB AO OB
  i j i j
  i j
  = +
  = + +
  = +
  atau
  
  1 5
  3 5
  4
  2
  | | | |
  = +
  | |
  
  \ . \ .
  | |
  =
  |
  \ .
  
  0
  6
  4
  2
  2 4 6
  A(1,3)
  B(5,5)
  i
  
  j
  
  O A
  B
  A B
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 2
  
  Exercise :
  
  Solutions :
  
  OA
  
  =
  
  OB =
  
  OC =
  
  OD =
  
  AB =
  
  BC =
  
  BC =
  
  CB =
  BD =
  
  DB =
  CD =
  
  DC
  
  Find the magnitude for each of the vectors
  Example :
  
  2 2
  3 2
  (3) ( 2)
  AB i j
  AB
  =
  = +
  
  13 = unit
  
  (a) 4 3 PQ i j = +
  (b) 12 5 a i j =
  
  (c) 8 2 b i j = +
  (d) 5 3 OA i j = +
  
  (e) 5 12 AB i j =
  B(5,2)
  2
  0
  4
  -2
  -4
  -4 -2 2 4
  A(-5,2)
  C(-3,-2)
  D(4,-5)
  B(5,2)
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 3
  
  (f) 6 8 OB i j =
  
  (g) 3 8 BO i j =
  
  Find the unit vector for each of the following
  Example :
  
  2 2
  3 2
  3 2
  (3) ( 2)
  3 2
  13
  AB i j
  i j
  AB
  i j
  =
  
  =
  +
  
  =
  
  (a) 2 6 a i j =
  
  2 6
  40
  i j
  
  (b) 8 6 PQ i j = +
  
  8 6
  10
  i j +
  
  (c) 12 5 r i j = +
  
  12 5
  13
  i j +
  
  (d)
  3
  12
  a
  | |
  =
  |
  \ .
  
  3 1
  12
  153
  | |
  |
  \ .
  
  (e)
  8
  7
  b
  | |
  =
  |
  
  \ .
  
  8 1
  7
  113
  | |
  |
  
  \ .
  
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 4
  
  Find the following vector
  Example :
  
  2 3 a i j =
  
  5 b i j = +
  Solution :
  
  2 2(2 3 ) 4 6 a i j i j = =
  
  3 3( 5 ) 3 15 b i j i j = + = +
  
  2 2(2 3 ) ( 5 )
  4 6 5
  3
  a b i j i j
  i j i j
  i j
  + = + +
  = +
  =
  
  Exercise :
  
  2 4 a i j =
  
  3 5 b i j = +
  
  2a =
  
  3b =
  
  2 a b =
  
  3 2 b a =
  
  2a b =
  
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 5
  
  EXERCISE
  1. Rajah menunjukkan dua vektor OA dan OB dengan O adalah asalan.
  
  Ungkapkan
  (a) OB dalam bentuk
  x
  y
  | |
  |
  \ .
  ,
  (b) dalam bentuk xi yj + ,
  (i) OA
  (ii) AB
  
  [(a)
  5
  3
  | |
  |
  \ .
  (b) (i) 8 4 i j + (b) 13i j ]
  
  2. Rajah menunjukkan dua vektor OP dan OQ.
  
  Ungkapkan
  (a) OQ dalam bentuk xi yj + ,
  (b) PQ dalam bentuk
  x
  y
  | |
  |
  \ .
  .
  
  [(a) 8 8 i j (b)
  4
  10
  | |
  |
  
  \ .
  ]
  
  y
  Q(8, -8)
  P(4, 2)
  O
  x
  y
  A(8, 4)
  B(5, 3)
  O
  x
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 6
  
  3. Diberi vektor 5 12 AB i j = . Cari
  (a) AB ,
  (b) vektor unit dalam arah AB.
  
  [(a) 13 (b)
  5 12
  13
  i j
  ]
  
  4. Diberi dua vektor 4 OP i j = + dan 5 7 OQ i j = + .
  (a) Ungkapkan vektor PQ dalam bentuk
  x
  y
  | |
  |
  \ .
  .
  (b) Vektor unit dalam arah PQ .
  
  [(a) 6 3 i j + (b)
  6 3
  45
  i j +
  ]
  
  5. Diberi O(0 , 0), A(7 , -4) dan B(2 , 8). Cari
  (a) AB dalam bentuk
  x
  y
  | |
  |
  \ .
  ,
  (b) nilai AB .
  
  [(a)
  5
  12
  | |
  |
  \ .
  (b) 13]
  
  INTERVENSI MATEMATIK TAMBAHAN 2012
  
  Panel Perunding Mata Pelajaran Matematik Tambahan, Jabatan Pelajaran Johor Page 7
  
  6. Diberi O(0 , 0), P(-4 , 8) dan Q(5 4). Cari
  (a) PQ dalam bentuk xi yj + ,
  (b) vektor unit dalam arah PQ .
  
  [(a) 9 12 i j (b)
  9 12
  15
  i j
  ]
  
  7. Diberi dua vektor 2 OA i j = dan 5 OB i j = + . Tentukan
  (a) AB ,
  (b) vektor unit dalam arah AB.
  
  [(a) 4 3 i j + (b)
  4 3
  5
  i j +
  ]
  
  8. Diberi bahawa 2 a i j = and 4 3 b i j = . Cari
  (a) vektor 2a b ,
  (b) vektor unit dalam arah 2a b .
  
  [(a) 2i j (b)
  2 7
  5
  i j
  ]
  
  ### Dapatkan aplikasi gratis kami
  
  Hak cipta © 2021 Scribd Inc.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a mathematics intervention module for Additional Mathematics from the Johor Education Department, focusing on the topic of Vectors. It includes notes, examples, and exercises with solutions. The document contains multiple questions and answers related to vector operations, such as expressing vectors in different forms, finding magnitudes, and unit vectors.
  
  Label: multiple-question-and-answer