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
  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

  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 → + 0 = 0 + 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 → ) = 0 {\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

  • 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
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

  • Transformasi
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