Tentukan Vektor Satuan Dari Vektor Vektor Berikut

Alumnice.co – Tentukan Vektor Satuan Dari Vektor Vektor Berikut

Dari Wikipedia bahasa Indonesia, ensiklopedia bebas

Vektor satuan
adalah suatu vektor yang ternormalisasi, yang berarti panjangnya bernilai 1. Umumnya dituliskan dalam menggunakan
topi
(bahasa Inggris:
Hat), sehingga:







u
^







{\displaystyle {\hat {u}}}




dibaca “u-topi” (‘u-hat’).

Suatu
vektor ternormalisasi







u
^







{\displaystyle {\hat {u}}}




dari suatu vektor
u
bernilai tidak nol, adalah suatu vektor yang berarah sama dengan
u, yaitu:








u
^





=



u






u







,


{\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\|\mathbf {u} \|}},}



di mana ||u|| adalah norma (atau panjang

atau besar) dari
u. Istilah
vektor ternormalisasi
kadang-kadang digunakan sebagai sinonim dari
vektor satuan. Dalam gaya penulisan yang lain (tidak menggunakan
huruf tebal) adalah dengan menggunakan panah di atas suatu variabel, yaitu








u
^





=




u












u












=




u





u


.


{\displaystyle {\hat {u}}={\frac {\vec {u}}{\|{\vec {u}}\|}}={\frac {\vec {u}}{u}}.}



Di sini








u








{\displaystyle \!{\vec {u}}}




adalah vektor yang dimaksud dan





u


{\displaystyle \!u}




adalah besarnya.

Vektor

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Posisi vektor

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a






=
(

a

1


,

a

2


)
=


(




a

1







a

2





)


=

a

1





i
^





+

a

2





j
^







{\displaystyle {\vec {a}}=(a_{1},a_{2})={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}}










a






=
(

a

1


,

a

2


,

a

3


)
=


(




a

1







a

2







a

3





)


=

a

1





i
^





+

a

2





j
^





+

a

3





k
^







{\displaystyle {\vec {a}}=(a_{1},a_{2},a_{3})={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}



Panjang vektor

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|
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Berada di





R

2




{\displaystyle R^{2}}



Panjang vektor a dalam posisi




(

a

1


,

a

2


)


{\displaystyle (a_{1},a_{2})}




adalah





|



a






|

=



a

1


2


+

a

2


2






{\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}}}}



Panjang vektor b dalam posisi




(

b

1


,

b

2


)


{\displaystyle (b_{1},b_{2})}




adalah





|



b






|

=



b

1


2


+

b

2


2






{\displaystyle \left|{\vec {b}}\right|={\sqrt {b_{1}^{2}+b_{2}^{2}}}}



Panjang vektor c dalam posisi




(

a

1


,

a

2


)


{\displaystyle (a_{1},a_{2})}




dan




(

b

1


,

b

2


)


{\displaystyle (b_{1},b_{2})}




adalah





|



c






|

=


(

b

1






a

1



)

2


+
(

b

2






a

2



)

2






{\displaystyle \left|{\vec {c}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}}}}



Berada di





R

3




{\displaystyle R^{3}}



Panjang vektor a dalam posisi




(

a

1


,

a

2


,

a

3


)


{\displaystyle (a_{1},a_{2},a_{3})}




adalah





|



a






|

=



a

1


2


+

a

2


2


+

a

3


2






{\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}}



Panjang vektor b dalam posisi




(

b

1


,

b

2


,

b

3


)


{\displaystyle (b_{1},b_{2},b_{3})}




adalah





|



b






|

=



b

1


2


+

b

2


2


+

b

3


2






{\displaystyle \left|{\vec {b}}\right|={\sqrt {b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}}}



Panjang vektor c dalam posisi




(

a

1


,

a

2


,

a

3


)


{\displaystyle (a_{1},a_{2},a_{3})}




dan




(

b

1


,

b

2


,

b

3


)


{\displaystyle (b_{1},b_{2},b_{3})}




adalah





|



c






|

=


(

b

1






a

1



)

2


+
(

b

2






a

2



)

2


+
(

b

3






a

3



)

2






{\displaystyle \left|{\vec {c}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+(b_{3}-a_{3})^{2}}}}



Vektor satuan

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a
^





=




a






|



a






|





{\displaystyle {\hat {a}}={\frac {\vec {a}}{\left|{\vec {a}}\right|}}}



Operasi aljabar pada vektor

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  • Penjumlahan dan pengurangan

terdiri dari 2 aturan jenis yaitu aturan segitiga dan jajar genjang








c






=



a






+



b






=


(




a

1







a

2





)


+


(




b

1







b

2





)


=


(





a

1


+

b

1









a

2


+

b

2






)




{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}+{\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}={\begin{pmatrix}{a_{1}+b_{1}}\\{a_{2}+b_{2}}\end{pmatrix}}}










c






=



a












b






=


(




a

1







a

2





)







(




b

1







b

2





)


=


(





a

1






b

1









a

2






b

2






)




{\displaystyle {\vec {c}}={\vec {a}}-{\vec {b}}={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}-{\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}={\begin{pmatrix}{a_{1}-b_{1}}\\{a_{2}-b_{2}}\end{pmatrix}}}



  • Perkalian
  1. skalar dengan vektor

Jika k skalar tak nol dan vektor







a






=

a

1





i
^





+

a

2





j
^





+

a

3





k
^







{\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}




maka vektor




k



a






=
(
k

a

1


,
k

a

2


,
k

a

3


)


{\displaystyle k{\vec {a}}=(ka_{1},ka_{2},ka_{3})}



  1. titik dua vektor

Jika vektor







a






=

a

1





i
^





+

a

2





j
^





+

a

3





k
^







{\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}




dan vektor







b






=

b

1





i
^





+

b

2





j
^





+

b

3





k
^







{\displaystyle {\vec {b}}=b_{1}{\hat {i}}+b_{2}{\hat {j}}+b_{3}{\hat {k}}}




maka







a












b






=

a

1



b

1


+

a

2



b

2


+

a

3



b

3




{\displaystyle {\vec {a}}\cdot {\vec {b}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}



  1. titik dua vektor dengan membentuk sudut

Jika







a








{\displaystyle {\vec {a}}}




dan







b








{\displaystyle {\vec {b}}}




vektor tak nol dan sudut




α




{\displaystyle \alpha }




diantara vektor







a








{\displaystyle {\vec {a}}}




dan







b








{\displaystyle {\vec {b}}}




maka perkalian skalar vektor







a








{\displaystyle {\vec {a}}}




dan







b








{\displaystyle {\vec {b}}}




adalah







a












b








{\displaystyle {\vec {a}}\cdot {\vec {b}}}




=





|



a






|





|



b






|

c
o
s
α




{\displaystyle \left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|cos\alpha }



  1. silang dua vektor

Jika vektor







a






=

a

1





i
^





+

a

2





j
^





+

a

3





k
^







{\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}




dan vektor







b






=

b

1





i
^





+

b

2





j
^





+

b

3





k
^







{\displaystyle {\vec {b}}=b_{1}{\hat {i}}+b_{2}{\hat {j}}+b_{3}{\hat {k}}}




maka







a






×





b






=
(

a

2



b

3





i
^





+

a

3



b

1





j
^





+

a

1



b

2





k
^





)



(

a

2



b

1





k
^





+

a

3



b

2





i
^





+

a

1



b

3





j
^





)


{\displaystyle {\vec {a}}\times {\vec {b}}=(a_{2}b_{3}{\hat {i}}+a_{3}b_{1}{\hat {j}}+a_{1}b_{2}{\hat {k}})-(a_{2}b_{1}{\hat {k}}+a_{3}b_{2}{\hat {i}}+a_{1}b_{3}{\hat {j}})}








[







i
^










j
^










k
^










i
^










j
^










a

1





a

2





a

3





a

1





a

2







b

1





b

2





b

3





b

1





b

2






]



{\displaystyle \left[{\begin{array}{rrr|rr}{\hat {i}}&{\hat {j}}&{\hat {k}}&{\hat {i}}&{\hat {j}}\\a_{1}&a_{2}&a_{3}&a_{1}&a_{2}\\b_{1}&b_{2}&b_{3}&b_{1}&b_{2}\\\end{array}}\right]}



  1. silang dua vektor dengan membentuk sudut

Jika







a








{\displaystyle {\vec {a}}}




dan







b








{\displaystyle {\vec {b}}}




vektor tak nol dan sudut




α




{\displaystyle \alpha }




diantara vektor







a








{\displaystyle {\vec {a}}}




dan







b








{\displaystyle {\vec {b}}}




maka perkalian skalar vektor







a








{\displaystyle {\vec {a}}}




dan







b








{\displaystyle {\vec {b}}}




adalah







a






×





b








{\displaystyle {\vec {a}}\times {\vec {b}}}




=





|



a






|

×



|



b






|

s
i
n
α




{\displaystyle \left|{\vec {a}}\right|\times \left|{\vec {b}}\right|sin\alpha }



Sifat operasi aljabar pada vektor

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|
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  1. a






    +



    b






    =



    b






    +



    a








    {\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}







  2. (



    a






    +



    b






    )
    +



    c






    =



    a






    +
    (



    b






    +



    c






    )


    {\displaystyle ({\vec {a}}+{\vec {b}})+{\vec {c}}={\vec {a}}+({\vec {b}}+{\vec {c}})}










  3. a






    +

    =

    +



    a








    {\displaystyle {\vec {a}}+0=0+{\vec {a}}}







  4. k
    (



    a






    +



    b






    )
    =
    k



    a






    +
    k



    b








    {\displaystyle k({\vec {a}}+{\vec {b}})=k{\vec {a}}+k{\vec {b}}}







  5. (
    k
    +
    l
    )



    a






    =
    k



    a






    +
    l



    a








    {\displaystyle (k+l){\vec {a}}=k{\vec {a}}+l{\vec {a}}}










  6. a






    +
    (






    a






    )
    =



    {\displaystyle {\vec {a}}+(-{\vec {a}})=0}










  7. a












    b






    =



    b












    a








    {\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}}







  8. (



    a












    b






    )






    c






    =



    a









    (



    b












    c






    )


    {\displaystyle ({\vec {a}}\cdot {\vec {b}})\cdot {\vec {c}}={\vec {a}}\cdot ({\vec {b}}\cdot {\vec {c}})}










  9. a









    1
    =
    1






    a








    {\displaystyle {\vec {a}}\cdot 1=1\cdot {\vec {a}}}







  10. k
    (



    a












    b






    )
    =
    k



    a












    b






    =



    a









    k



    b








    {\displaystyle k({\vec {a}}\cdot {\vec {b}})=k{\vec {a}}\cdot {\vec {b}}={\vec {a}}\cdot k{\vec {b}}}







  11. (
    k



    l
    )



    a






    =
    k
    (
    l






    a






    )


    {\displaystyle (k\cdot l){\vec {a}}=k(l\cdot {\vec {a}})}










  12. a












    a






    =


    |



    a






    |


    2




    {\displaystyle {\vec {a}}\cdot {\vec {a}}=\left|{\vec {a}}\right|^{2}}










  13. a






    ×





    b












    b






    ×





    a








    {\displaystyle {\vec {a}}\times {\vec {b}}\neq {\vec {b}}\times {\vec {a}}}










  14. a






    ×





    b






    =



    (



    b






    ×





    a






    )


    {\displaystyle {\vec {a}}\times {\vec {b}}=-({\vec {b}}\times {\vec {a}})}







  15. (



    a






    ×





    b






    )
    ×





    c












    a






    ×


    (



    b






    ×





    c






    )


    {\displaystyle ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\neq {\vec {a}}\times ({\vec {b}}\times {\vec {c}})}










  16. a









    (



    b






    ×





    c






    )
    =



    b









    (



    c






    ×





    a






    )
    =



    c









    (



    a






    ×





    b






    )


    {\displaystyle {\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})={\vec {b}}\cdot ({\vec {c}}\times {\vec {a}})={\vec {c}}\cdot ({\vec {a}}\times {\vec {b}})}










  17. a






    ×


    (



    b






    +



    c






    )
    =



    a






    ×





    b






    +



    a






    ×





    c








    {\displaystyle {\vec {a}}\times ({\vec {b}}+{\vec {c}})={\vec {a}}\times {\vec {b}}+{\vec {a}}\times {\vec {c}}}







  18. k
    (



    a






    ×





    b






    )
    =
    k



    a






    ×





    b






    =



    a






    ×


    k



    b








    {\displaystyle k({\vec {a}}\times {\vec {b}})=k{\vec {a}}\times {\vec {b}}={\vec {a}}\times k{\vec {b}}}



Hubungan vektor dengan vektor lain

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  • Perkalian titik
Saling tegak lurus

Jika tegak lurus antara vektor







a








{\displaystyle {\vec {a}}}




dengan vektor







b








{\displaystyle {\vec {b}}}




maka








a












b






=

|



a






|





|



b






|

cos





90









{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cos {90}^{\circ }}










a












b






=



{\displaystyle {\vec {a}}\cdot {\vec {b}}=0}



Sejajar

Jika vektor







a








{\displaystyle {\vec {a}}}




sejajar dengan vektor







b








{\displaystyle {\vec {b}}}




maka








a












b






=

|



a






|





|



b






|

cos















{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cos {0}^{\circ }}










a












b






=

|



a






|





|



b






|



{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}










a












b






=

|



a






|





|



b






|

cos





180









{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cos {180}^{\circ }}










a












b






=




|



a






|





|



b






|



{\displaystyle {\vec {a}}\cdot {\vec {b}}=-\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}



  • Perkalian silang
Saling tegak lurus

Jika tegak lurus antara vektor







a








{\displaystyle {\vec {a}}}




dengan vektor







b








{\displaystyle {\vec {b}}}




maka








a






×





b






=

|



a






|





|



b






|

sin





90









{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\sin {90}^{\circ }}










a






×





b






=

|



a






|





|



b






|



{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}










a






×





b






=

|



a






|





|



b






|

sin





270









{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\sin {270}^{\circ }}










a






×





b






=




|



a






|





|



b






|



{\displaystyle {\vec {a}}\times {\vec {b}}=-\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}



Jika



β


>


90









{\displaystyle \beta >{90}^{\circ }}






β


<


90









{\displaystyle \beta <{90}^{\circ }}




maka vektor saling berlawanan arah

Sejajar

Jika vektor







a








{\displaystyle {\vec {a}}}




sejajar dengan vektor







b








{\displaystyle {\vec {b}}}




maka








a






×





b






=

|



a






|





|



b






|

sin















{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\sin {0}^{\circ }}










a






×





b






=



{\displaystyle {\vec {a}}\times {\vec {b}}=0}



Sudut dua vektor

[sunting
|
sunting sumber]

Jika vektor







a








{\displaystyle {\vec {a}}}




dan vektor







b








{\displaystyle {\vec {b}}}




sudut yang dapat dibentuk dari kedua vektor tersebut adalah




c
o
s
α


=






a












b









|



a






|





|



b






|






{\displaystyle cos\alpha ={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}}}



Panjang proyeksi dan proyeksi vektor

[sunting
|
sunting sumber]

Panjang proyeksi vektor







a








{\displaystyle {\vec {a}}}




pada vektor







b








{\displaystyle {\vec {b}}}




adalah





|



c






|

=






a












b








|



b






|





{\displaystyle \left|{\vec {c}}\right|={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|}}}



Proyeksi vektor







a








{\displaystyle {\vec {a}}}




pada vektor







b








{\displaystyle {\vec {b}}}




adalah







c






=






a












b









|



b






|


2










b








{\displaystyle {\vec {c}}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|^{2}}}\cdot {\vec {b}}}



Perbandingan

[sunting
|
sunting sumber]

Aturan jajar genjang
Posisi vektor







N






=



m
s
+
n
r


m
+
n





{\displaystyle {\vec {N}}={\frac {ms+nr}{m+n}}}



Berada di





R

2




{\displaystyle R^{2}}










N






=
(



m

x

2


+
n

x

1




m
+
n



,



m

y

2


+
n

y

1




m
+
n



)


{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}})}



Berada di





R

3




{\displaystyle R^{3}}










N






=
(



m

x

2


+
n

x

1




m
+
n



,



m

y

2


+
n

y

1




m
+
n



,



m

z

2


+
n

z

1




m
+
n



)


{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}},{\frac {mz_{2}+nz_{1}}{m+n}})}



Satu garis
  • Perbandingan posisi dalam adalah m:n
Posisi vektor








N






=



m
s
+
n
r


m
+
n





{\displaystyle {\vec {N}}={\frac {ms+nr}{m+n}}}



Berada di





R

2




{\displaystyle R^{2}}










N






=
(



m

x

2


+
n

x

1




m
+
n



,



m

y

2


+
n

y

1




m
+
n



)


{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}})}



Berada di





R

3




{\displaystyle R^{3}}










N






=
(



m

x

2


+
n

x

1




m
+
n



,



m

y

2


+
n

y

1




m
+
n



,



m

z

2


+
n

z

1




m
+
n



)


{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}},{\frac {mz_{2}+nz_{1}}{m+n}})}



  • Perbandingan posisi luar adalah m:-n
Posisi vektor








N






=



m
s



n
r


m



n





{\displaystyle {\vec {N}}={\frac {ms-nr}{m-n}}}



Berada di





R

2




{\displaystyle R^{2}}










N






=
(



m

x

2





n

x

1




m



n



,



m

y

2





n

y

1




m



n



)


{\displaystyle {\vec {N}}=({\frac {mx_{2}-nx_{1}}{m-n}},{\frac {my_{2}-ny_{1}}{m-n}})}



Berada di





R

3




{\displaystyle R^{3}}










N






=
(



m

x

2





n

x

1




m



n



,



m

y

2





n

y

1




m



n



,



m

z

2





n

z

1




m



n



)


{\displaystyle {\vec {N}}=({\frac {mx_{2}-nx_{1}}{m-n}},{\frac {my_{2}-ny_{1}}{m-n}},{\frac {mz_{2}-nz_{1}}{m-n}})}



Transformasi

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Transformasi terdiri dari 2 jenis yaitu:

  • Transformasi isometri

Transformasi isometri adalah transformasi yang dapat mengubah bentuknya. Contohnya translasi (penggeseran), refleksi (perpindahan) dan rotasi (perputaran).

  • Transformasi nonisometri

Transformasi nonisometri adalah transformasi yang tidak dapat mengubah bentuknya. Contohnya dilatasi (perubahan), stretching (regangan) dan shearing (gusuran).

Translasi

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Rumus translasi adalah:






(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}




+






(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



Refleksi

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|
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Rumus refleksi adalah:

tanpa titik pusat







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



c
o
s
2
α




s
i
n
2
α






s
i
n
2
α







c
o
s
2
α





)




{\displaystyle {\begin{pmatrix}cos2\alpha &sin2\alpha \\sin2\alpha &-cos2\alpha \end{pmatrix}}}










(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



dengan titik pusat (a,b)







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



c
o
s
2
α




s
i
n
2
α






s
i
n
2
α







c
o
s
2
α





)




{\displaystyle {\begin{pmatrix}cos2\alpha &sin2\alpha \\sin2\alpha &-cos2\alpha \end{pmatrix}}}










(



x



a




y



b



)




{\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}




+






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}



Rotasi

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|
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Rumus rotasi adalah:

tanpa titik pusat







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



c
o
s
α







s
i
n
α






s
i
n
α




c
o
s
α





)




{\displaystyle {\begin{pmatrix}cos\alpha &-sin\alpha \\sin\alpha &cos\alpha \end{pmatrix}}}










(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



dengan titik pusat (a,b)







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



c
o
s
α







s
i
n
α






s
i
n
α




c
o
s
α





)




{\displaystyle {\begin{pmatrix}cos\alpha &-sin\alpha \\sin\alpha &cos\alpha \end{pmatrix}}}










(



x



a




y



b



)




{\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}




+






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}



Dilatasi

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Rumus dilatasi adalah:

tanpa titik pusat







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



k










k



)




{\displaystyle {\begin{pmatrix}k&0\\0&k\end{pmatrix}}}










(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



dengan titik pusat (a,b)







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



k










k



)




{\displaystyle {\begin{pmatrix}k&0\\0&k\end{pmatrix}}}










(



x



a




y



b



)




{\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}




+






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}



Stretching

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Rumus stretching adalah:

sumbu x

tanpa titik pusat







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



k










1



)




{\displaystyle {\begin{pmatrix}k&0\\0&1\end{pmatrix}}}










(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



dengan titik pusat (a,b)







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



k










1



)




{\displaystyle {\begin{pmatrix}k&0\\0&1\end{pmatrix}}}










(



x



a




y



b



)




{\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}




+






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}



sumbu y

tanpa titik pusat







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



1










k



)




{\displaystyle {\begin{pmatrix}1&0\\0&k\end{pmatrix}}}










(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



dengan titik pusat (a,b)







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



1










k



)




{\displaystyle {\begin{pmatrix}1&0\\0&k\end{pmatrix}}}










(



x



a




y



b



)




{\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}




+






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}



Shearing

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|
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Rumus shearing adalah:

sumbu x

tanpa titik pusat







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



1


k







1



)




{\displaystyle {\begin{pmatrix}1&k\\0&1\end{pmatrix}}}










(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



dengan titik pusat (a,b)







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



1


k







1



)




{\displaystyle {\begin{pmatrix}1&k\\0&1\end{pmatrix}}}










(



x



a




y



b



)




{\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}




+






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}



sumbu y

tanpa titik pusat







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



1







k


1



)




{\displaystyle {\begin{pmatrix}1&0\\k&1\end{pmatrix}}}










(



x




y



)




{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}



dengan titik pusat (a,b)







(




x







y





)




{\displaystyle {\begin{pmatrix}x’\\y’\end{pmatrix}}}




=






(



1







k


1



)




{\displaystyle {\begin{pmatrix}1&0\\k&1\end{pmatrix}}}










(



x



a




y



b



)




{\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}




+






(



a




b



)




{\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}



Rumus sederhana
Keterangan Posisi Hasil
Translasi
penggeseran (a,b)




(
x
,
y
)


{\displaystyle (x,y)}







(
x
+
a
,
y
+
b
)


{\displaystyle (x+a,y+b)}



Refleksi
sumbu x [0°]




(
x
,
y
)


{\displaystyle (x,y)}







(
x
,



y
)


{\displaystyle (x,-y)}



sumbu y [90°]




(
x
,
y
)


{\displaystyle (x,y)}







(



x
,
y
)


{\displaystyle (-x,y)}



y=x [45°]




(
x
,
y
)


{\displaystyle (x,y)}







(
y
,
x
)


{\displaystyle (y,x)}



y=-x [135°]




(
x
,
y
)


{\displaystyle (x,y)}







(



y
,



x
)


{\displaystyle (-y,-x)}



pusat (0,0) [0° dan 90°]




(
x
,
y
)


{\displaystyle (x,y)}







(



x
,



y
)


{\displaystyle (-x,-y)}



pusat (a,b) [0° dan 90°]




(
x
,
y
)


{\displaystyle (x,y)}







(
2
a



x
,
2
b



y
)


{\displaystyle (2a-x,2b-y)}



pusat (a,0) [0° dan 90°]




(
x
,
y
)


{\displaystyle (x,y)}







(
2
a



x
,
y
)


{\displaystyle (2a-x,y)}



pusat (0,b) [0° dan 90°]




(
x
,
y
)


{\displaystyle (x,y)}







(
x
,
2
b



y
)


{\displaystyle (x,2b-y)}



Rotasi
berpusat (0,0)
90°




(
x
,
y
)


{\displaystyle (x,y)}







(



y
,
x
)


{\displaystyle (-y,x)}



-90°




(
x
,
y
)


{\displaystyle (x,y)}







(
y
,



x
)


{\displaystyle (y,-x)}



180°




(
x
,
y
)


{\displaystyle (x,y)}







(



x
,



y
)


{\displaystyle (-x,-y)}



berpusat (a,b)
90°




(
x
,
y
)


{\displaystyle (x,y)}







(



y
+
a
+
b
,
x



a
+
b
)


{\displaystyle (-y+a+b,x-a+b)}



-90°




(
x
,
y
)


{\displaystyle (x,y)}







(
y



a
+
b
,



x
+
a
+
b
)


{\displaystyle (y-a+b,-x+a+b)}



180°




(
x
,
y
)


{\displaystyle (x,y)}







(



x
+
2
a
,



y
+
2
b
)


{\displaystyle (-x+2a,-y+2b)}



berpusat (0,0)
Dilatasi
skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
k



x
,
k



y
)


{\displaystyle (k\cdot x,k\cdot y)}



Stretching
sumbu x dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
k



x
,
y
)


{\displaystyle (k\cdot x,y)}



sumbu y dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
x
,
k



y
)


{\displaystyle (x,k\cdot y)}



Shearing
sumbu x dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
k



x
+
y
,
y
)


{\displaystyle (k\cdot x+y,y)}



sumbu y dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
x
,
x
+
k



y
)


{\displaystyle (x,x+k\cdot y)}



berpusat (a,b)
Dilatasi
skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
k



x
+
(
1



k
)
a
,
k



y
+
(
1



k
)
b
)


{\displaystyle (k\cdot x+(1-k)a,k\cdot y+(1-k)b)}



Stretching
sumbu x dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
k



x
+
(
1



k
)
a
,
y
)


{\displaystyle (k\cdot x+(1-k)a,y)}



sumbu y dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
x
,
k



y
+
(
1



k
)
b
)


{\displaystyle (x,k\cdot y+(1-k)b)}



Shearing
sumbu x dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
x
+
k



(
y



b
)
)
,
y
)


{\displaystyle (x+k\cdot (y-b)),y)}



sumbu y dan skala k




(
x
,
y
)


{\displaystyle (x,y)}







(
x
,
y
+
k



(
x



a
)
)


{\displaystyle (x,y+k\cdot (x-a))}



Lihat pula

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  • Transformasi



Tentukan Vektor Satuan Dari Vektor Vektor Berikut

Sumber: https://id.wikipedia.org/wiki/Vektor_satuan

Baca :   Contoh Soal Sejarah Kelas 10 Semester 2

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