Bagian ini memerlukan
pengembangan . Anda dapat membantu dengan
mengembangkannya .
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}}}
Suatu vektor ternormalisasi
u
^ ^ -->
{\displaystyle {\hat {u}}}
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}}}
u
{\displaystyle \!u}
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})}
|
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})}
|
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})}
(
b
1
,
b
2
)
{\displaystyle (b_{1},b_{2})}
|
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})}
|
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})}
|
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})}
(
b
1
,
b
2
,
b
3
)
{\displaystyle (b_{1},b_{2},b_{3})}
|
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}}}
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}}}
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}}}
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}}}
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}}}
b
→ → -->
{\displaystyle {\vec {b}}}
α α -->
{\displaystyle \alpha }
a
→ → -->
{\displaystyle {\vec {a}}}
b
→ → -->
{\displaystyle {\vec {b}}}
a
→ → -->
{\displaystyle {\vec {a}}}
b
→ → -->
{\displaystyle {\vec {b}}}
a
→ → -->
⋅ ⋅ -->
b
→ → -->
{\displaystyle {\vec {a}}\cdot {\vec {b}}}
|
a
→ → -->
|
⋅ ⋅ -->
|
b
→ → -->
|
c
o
s
α 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}}}
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}}}
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}}}
b
→ → -->
{\displaystyle {\vec {b}}}
α α -->
{\displaystyle \alpha }
a
→ → -->
{\displaystyle {\vec {a}}}
b
→ → -->
{\displaystyle {\vec {b}}}
a
→ → -->
{\displaystyle {\vec {a}}}
b
→ → -->
{\displaystyle {\vec {b}}}
a
→ → -->
× × -->
b
→ → -->
{\displaystyle {\vec {a}}\times {\vec {b}}}
|
a
→ → -->
|
× × -->
|
b
→ → -->
|
s
i
n
α 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
Saling tegak lurus Jika tegak lurus antara vektor
a
→ → -->
{\displaystyle {\vec {a}}}
b
→ → -->
{\displaystyle {\vec {b}}}
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}}}
b
→ → -->
{\displaystyle {\vec {b}}}
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|}
Saling tegak lurus Jika tegak lurus antara vektor
a
→ → -->
{\displaystyle {\vec {a}}}
b
→ → -->
{\displaystyle {\vec {b}}}
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 β -->
>
90
∘ ∘ -->
{\displaystyle \beta >{90}^{\circ }}
Jika
β β -->
<
90
∘ ∘ -->
{\displaystyle \beta <{90}^{\circ }}
Sejajar Jika vektor
a
→ → -->
{\displaystyle {\vec {a}}}
b
→ → -->
{\displaystyle {\vec {b}}}
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}}}
b
→ → -->
{\displaystyle {\vec {b}}}
c
o
s
α 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}}}
b
→ → -->
{\displaystyle {\vec {b}}}
|
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}}}
b
→ → -->
{\displaystyle {\vec {b}}}
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 terdiri dari 2 jenis yaitu:
Transformasi isometri adalah transformasi yang dapat mengubah bentuknya. Contohnya translasi (penggeseran), refleksi (perpindahan) dan rotasi (perputaran).
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 α -->
s
i
n
2
α α -->
s
i
n
2
α α -->
− − -->
c
o
s
2
α 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 α -->
s
i
n
2
α α -->
s
i
n
2
α α -->
− − -->
c
o
s
2
α 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 α -->
− − -->
s
i
n
α α -->
s
i
n
α α -->
c
o
s
α 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 α -->
− − -->
s
i
n
α α -->
s
i
n
α α -->
c
o
s
α 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
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
![[icon]](Berkas:Wiki_letter_w_cropped.svg){kind=small}
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4452fd5552ab9df872f08828c23d4f169ee15570)
