- Source: Vektor satuan
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
= Posisi vektor
=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
=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}}}}
Jumlah dan selisih kedua vektor
|
a
→
±
b
→
|
=
|
a
→
|
2
+
|
b
→
|
2
±
2
a
→
⋅
b
→
⋅
c
o
s
C
{\displaystyle \left|{\vec {a}}\pm {\vec {b}}\right|={\sqrt {|{\vec {a}}|^{2}+|{\vec {b}}|^{2}\pm 2{\vec {a}}\cdot {\vec {b}}\cdot cosC}}}
= Vektor satuan
=a
^
=
a
→
|
a
→
|
{\displaystyle {\hat {a}}={\frac {\vec {a}}{\left|{\vec {a}}\right|}}}
= Operasi aljabar pada vektor
=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
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})}
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}}
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 }
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]}
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
=a
→
+
b
→
=
b
→
+
a
→
{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}
(
a
→
+
b
→
)
+
c
→
=
a
→
+
(
b
→
+
c
→
)
{\displaystyle ({\vec {a}}+{\vec {b}})+{\vec {c}}={\vec {a}}+({\vec {b}}+{\vec {c}})}
a
→
+
0
=
0
+
a
→
{\displaystyle {\vec {a}}+0=0+{\vec {a}}}
k
(
a
→
+
b
→
)
=
k
a
→
+
k
b
→
{\displaystyle k({\vec {a}}+{\vec {b}})=k{\vec {a}}+k{\vec {b}}}
(
k
+
l
)
a
→
=
k
a
→
+
l
a
→
{\displaystyle (k+l){\vec {a}}=k{\vec {a}}+l{\vec {a}}}
a
→
+
(
−
a
→
)
=
0
{\displaystyle {\vec {a}}+(-{\vec {a}})=0}
a
→
⋅
b
→
=
b
→
⋅
a
→
{\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}}
(
a
→
⋅
b
→
)
⋅
c
→
=
a
→
⋅
(
b
→
⋅
c
→
)
{\displaystyle ({\vec {a}}\cdot {\vec {b}})\cdot {\vec {c}}={\vec {a}}\cdot ({\vec {b}}\cdot {\vec {c}})}
a
→
⋅
1
=
1
⋅
a
→
{\displaystyle {\vec {a}}\cdot 1=1\cdot {\vec {a}}}
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}}}
(
k
⋅
l
)
a
→
=
k
(
l
⋅
a
→
)
{\displaystyle (k\cdot l){\vec {a}}=k(l\cdot {\vec {a}})}
a
→
⋅
a
→
=
|
a
→
|
2
{\displaystyle {\vec {a}}\cdot {\vec {a}}=\left|{\vec {a}}\right|^{2}}
a
→
×
b
→
≠
b
→
×
a
→
{\displaystyle {\vec {a}}\times {\vec {b}}\neq {\vec {b}}\times {\vec {a}}}
a
→
×
b
→
=
−
(
b
→
×
a
→
)
{\displaystyle {\vec {a}}\times {\vec {b}}=-({\vec {b}}\times {\vec {a}})}
(
a
→
×
b
→
)
×
c
→
≠
a
→
×
(
b
→
×
c
→
)
{\displaystyle ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\neq {\vec {a}}\times ({\vec {b}}\times {\vec {c}})}
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}})}
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}}}
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
=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
→
=
0
{\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
0
∘
{\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 }}
maka dua vektor tersebut searah
Jika
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
0
∘
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\sin {0}^{\circ }}
a
→
×
b
→
=
0
{\displaystyle {\vec {a}}\times {\vec {b}}=0}
= Sudut dua vektor
=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
=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}}}
= Metode
=segitiga
R
→
=
a
→
+
b
→
{\displaystyle {\vec {R}}={\vec {a}}+{\vec {b}}}
jajar genjang
R
→
=
|
a
→
−
b
→
|
=
|
a
→
|
2
+
|
b
→
|
2
−
2
⋅
a
→
⋅
b
→
⋅
c
o
s
C
{\displaystyle {\vec {R}}=|{\vec {a}}-{\vec {b}}|={\sqrt {|{\vec {a}}|^{2}+|{\vec {b}}|^{2}-2\cdot {\vec {a}}\cdot {\vec {b}}\cdot cosC}}}
= Perbandingan
=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
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
=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
=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
=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
=Rumus dilatasi adalah:
tanpa titik pusat
(
x
′
y
′
)
{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}
=
(
k
0
0
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
0
0
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
=Rumus stretching adalah:
sumbu x
tanpa titik pusat
(
x
′
y
′
)
{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}
=
(
k
0
0
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
0
0
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
0
0
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
0
0
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
=Rumus shearing adalah:
sumbu x
tanpa titik pusat
(
x
′
y
′
)
{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}
=
(
1
k
0
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
0
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
0
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
0
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
Lihat pula
Transformasi
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