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 . Isitilah 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}
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}}}
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}}}}
a
^
=
a
→
|
a
→
|
{\displaystyle {\hat {a}}={\frac {\vec {a}}{\left|{\vec {a}}\right|}}}
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
{\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
{\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}})}
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
{\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
{\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]}
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
→
|
{\displaystyle {\vec {a}}\cdot {\vec {a}}=\left|{\vec {a}}\right|}
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}}}
Saling tegak lurus Jika tegak lurus antara vektor
a
→
{\displaystyle {\vec {a}}}
b
→
{\displaystyle {\vec {b}}}
a
→
⋅
b
→
=
|
a
→
|
⋅
|
b
→
|
c
o
s
90
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|cos90}
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
→
|
c
o
s
0
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|cos0}
a
→
⋅
b
→
=
|
a
→
|
⋅
|
b
→
|
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}
a
→
⋅
b
→
=
|
a
→
|
⋅
|
b
→
|
c
o
s
180
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|cos180}
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
→
|
s
i
n
90
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|sin90}
a
→
×
b
→
=
|
a
→
|
⋅
|
b
→
|
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}
a
→
×
b
→
=
|
a
→
|
⋅
|
b
→
|
s
i
n
180
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|sin180}
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}
Jika
β
<
90
{\displaystyle \beta <90}
Sejajar Jika vektor
a
→
{\displaystyle {\vec {a}}}
b
→
{\displaystyle {\vec {b}}}
a
→
×
b
→
=
|
a
→
|
⋅
|
b
→
|
s
i
n
0
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|sin0}
a
→
×
b
→
=
0
{\displaystyle {\vec {a}}\times {\vec {b}}=0}
Jika vektor
a
→
{\displaystyle {\vec {a}}}
b
→
{\displaystyle {\vec {b}}}
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 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}}}
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
Perbandingan posisi luar adalah m:-n 
Artikel utama untuk bagian ini adalah: 
![{\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)
