Posisi vektor

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

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

Panjang vektor

Berada di ${\displaystyle R^{2}}$

Panjang vektor a dalam posisi ${\displaystyle (a_{1},a_{2})}$ adalah ${\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}}}}$

Panjang vektor b dalam posisi ${\displaystyle (b_{1},b_{2})}$ adalah ${\displaystyle \left|{\vec {b}}\right|={\sqrt {b_{1}^{2}+b_{2}^{2}}}}$

Panjang vektor c dalam posisi ${\displaystyle (a_{1},a_{2})}$ dan ${\displaystyle (b_{1},b_{2})}$ adalah ${\displaystyle \left|{\vec {c}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}}}}$

Berada di ${\displaystyle R^{3}}$

Panjang vektor a dalam posisi ${\displaystyle (a_{1},a_{2},a_{3})}$ adalah ${\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}}$

Panjang vektor b dalam posisi ${\displaystyle (b_{1},b_{2},b_{3})}$ adalah ${\displaystyle \left|{\vec {b}}\right|={\sqrt {b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}}}$

Panjang vektor c dalam posisi ${\displaystyle (a_{1},a_{2},a_{3})}$ dan ${\displaystyle (b_{1},b_{2},b_{3})}$ adalah ${\displaystyle \left|{\vec {c}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+(b_{3}-a_{3})^{2}}}}$

Jumlah dan selisih kedua vektor

${\displaystyle \left|{\vec {a}}\pm {\vec {b}}\right|={\sqrt {|{\vec {a}}|^{2}+|{\vec {b}}|^{2}\pm 2{\vec {a}}\cdot {\vec {b}}\cdot cosC}}}$

Vektor satuan

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

Operasi aljabar pada vektor

• Penjumlahan dan pengurangan

terdiri dari 2 aturan jenis yaitu aturan segitiga dan jajar genjang

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

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

• Perkalian

1. skalar dengan vektor

Jika k skalar tak nol dan vektor ${\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}$ maka vektor ${\displaystyle k{\vec {a}}=(ka_{1},ka_{2},ka_{3})}$

1. titik dua vektor

Jika vektor ${\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}$ dan vektor ${\displaystyle {\vec {b}}=b_{1}{\hat {i}}+b_{2}{\hat {j}}+b_{3}{\hat {k}}}$ maka ${\displaystyle {\vec {a}}\cdot {\vec {b}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}$

1. titik dua vektor dengan membentuk sudut

Jika ${\displaystyle {\vec {a}}}$ dan ${\displaystyle {\vec {b}}}$ vektor tak nol dan sudut ${\displaystyle \alpha }$ di antara vektor ${\displaystyle {\vec {a}}}$ dan ${\displaystyle {\vec {b}}}$ maka perkalian skalar vektor ${\displaystyle {\vec {a}}}$ dan ${\displaystyle {\vec {b}}}$ adalah ${\displaystyle {\vec {a}}\cdot {\vec {b}}}$ = ${\displaystyle \left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|cos\alpha }$

1. silang dua vektor

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

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

1. silang dua vektor dengan membentuk sudut

Jika ${\displaystyle {\vec {a}}}$ dan ${\displaystyle {\vec {b}}}$ vektor tak nol dan sudut ${\displaystyle \alpha }$ di antara vektor ${\displaystyle {\vec {a}}}$ dan ${\displaystyle {\vec {b}}}$ maka perkalian skalar vektor ${\displaystyle {\vec {a}}}$ dan ${\displaystyle {\vec {b}}}$ adalah ${\displaystyle {\vec {a}}\times {\vec {b}}}$ = ${\displaystyle \left|{\vec {a}}\right|\times \left|{\vec {b}}\right|sin\alpha }$

Sifat operasi aljabar pada vektor

1. ${\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}$
2. ${\displaystyle ({\vec {a}}+{\vec {b}})+{\vec {c}}={\vec {a}}+({\vec {b}}+{\vec {c}})}$
3. ${\displaystyle {\vec {a}}+0=0+{\vec {a}}}$
4. ${\displaystyle k({\vec {a}}+{\vec {b}})=k{\vec {a}}+k{\vec {b}}}$
5. ${\displaystyle (k+l){\vec {a}}=k{\vec {a}}+l{\vec {a}}}$
6. ${\displaystyle {\vec {a}}+(-{\vec {a}})=0}$
7. ${\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}}$
8. ${\displaystyle ({\vec {a}}\cdot {\vec {b}})\cdot {\vec {c}}={\vec {a}}\cdot ({\vec {b}}\cdot {\vec {c}})}$
9. ${\displaystyle {\vec {a}}\cdot 1=1\cdot {\vec {a}}}$
10. ${\displaystyle k({\vec {a}}\cdot {\vec {b}})=k{\vec {a}}\cdot {\vec {b}}={\vec {a}}\cdot k{\vec {b}}}$
11. ${\displaystyle (k\cdot l){\vec {a}}=k(l\cdot {\vec {a}})}$
12. ${\displaystyle {\vec {a}}\cdot {\vec {a}}=\left|{\vec {a}}\right|^{2}}$
13. ${\displaystyle {\vec {a}}\times {\vec {b}}\neq {\vec {b}}\times {\vec {a}}}$
14. ${\displaystyle {\vec {a}}\times {\vec {b}}=-({\vec {b}}\times {\vec {a}})}$
15. ${\displaystyle ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\neq {\vec {a}}\times ({\vec {b}}\times {\vec {c}})}$
16. ${\displaystyle {\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})={\vec {b}}\cdot ({\vec {c}}\times {\vec {a}})={\vec {c}}\cdot ({\vec {a}}\times {\vec {b}})}$
17. ${\displaystyle {\vec {a}}\times ({\vec {b}}+{\vec {c}})={\vec {a}}\times {\vec {b}}+{\vec {a}}\times {\vec {c}}}$
18. ${\displaystyle k({\vec {a}}\times {\vec {b}})=k{\vec {a}}\times {\vec {b}}={\vec {a}}\times k{\vec {b}}}$

Hubungan vektor dengan vektor lain

• Perkalian titik

Saling tegak lurus

Jika tegak lurus antara vektor ${\displaystyle {\vec {a}}}$ dengan vektor ${\displaystyle {\vec {b}}}$ maka

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

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

Sejajar

Jika vektor ${\displaystyle {\vec {a}}}$ sejajar dengan vektor ${\displaystyle {\vec {b}}}$ maka

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

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

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

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

• Perkalian silang

Saling tegak lurus

Jika tegak lurus antara vektor ${\displaystyle {\vec {a}}}$ dengan vektor ${\displaystyle {\vec {b}}}$ maka

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

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

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

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

Jika ${\displaystyle \beta >{90}^{\circ }}$ maka dua vektor tersebut searah

Jika ${\displaystyle \beta <{90}^{\circ }}$ maka vektor saling berlawanan arah

Sejajar

Jika vektor ${\displaystyle {\vec {a}}}$ sejajar dengan vektor ${\displaystyle {\vec {b}}}$ maka

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

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

Sudut dua vektor

Jika vektor ${\displaystyle {\vec {a}}}$ dan vektor ${\displaystyle {\vec {b}}}$ sudut yang dapat dibentuk dari kedua vektor tersebut adalah ${\displaystyle cos\alpha ={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}}}$

Panjang proyeksi dan proyeksi vektor

Panjang proyeksi vektor ${\displaystyle {\vec {a}}}$ pada vektor ${\displaystyle {\vec {b}}}$ adalah ${\displaystyle \left|{\vec {c}}\right|={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|}}}$

Proyeksi vektor ${\displaystyle {\vec {a}}}$ pada vektor ${\displaystyle {\vec {b}}}$ adalah ${\displaystyle {\vec {c}}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|^{2}}}\cdot {\vec {b}}}$

Metode

segitiga

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

jajar genjang

${\displaystyle {\vec {R}}=|{\vec {a}}-{\vec {b}}|={\sqrt {|{\vec {a}}|^{2}+|{\vec {b}}|^{2}-2\cdot {\vec {a}}\cdot {\vec {b}}\cdot cosC}}}$

Perbandingan

Aturan jajar genjang

Posisi vektor

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

Berada di ${\displaystyle R^{2}}$

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

Berada di ${\displaystyle R^{3}}$

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

Satu garis

• Perbandingan posisi dalam adalah m:n

Posisi vektor

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

Berada di ${\displaystyle R^{2}}$

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

Berada di ${\displaystyle R^{3}}$

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

• Perbandingan posisi luar adalah m:-n

Posisi vektor

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

Berada di ${\displaystyle R^{2}}$

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

Berada di ${\displaystyle R^{3}}$

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

Translasi

Rumus translasi adalah: ${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

Refleksi

Rumus refleksi adalah:

tanpa titik pusat

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}cos2\alpha &sin2\alpha \\sin2\alpha &-cos2\alpha \end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

dengan titik pusat (a,b)

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}cos2\alpha &sin2\alpha \\sin2\alpha &-cos2\alpha \end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$

Rotasi

Rumus rotasi adalah:

tanpa titik pusat

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}cos\alpha &-sin\alpha \\sin\alpha &cos\alpha \end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

dengan titik pusat (a,b)

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}cos\alpha &-sin\alpha \\sin\alpha &cos\alpha \end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$

Dilatasi

Rumus dilatasi adalah:

tanpa titik pusat

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}k&0\\0&k\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

dengan titik pusat (a,b)

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}k&0\\0&k\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$

Stretching

Rumus stretching adalah:

sumbu x

tanpa titik pusat

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}k&0\\0&1\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

dengan titik pusat (a,b)

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}k&0\\0&1\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$

sumbu y

tanpa titik pusat

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}1&0\\0&k\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

dengan titik pusat (a,b)

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}1&0\\0&k\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$

Shearing

Rumus shearing adalah:

sumbu x

tanpa titik pusat

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}1&k\\0&1\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

dengan titik pusat (a,b)

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}1&k\\0&1\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$

sumbu y

tanpa titik pusat

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}1&0\\k&1\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}}$

dengan titik pusat (a,b)

${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}}$ = ${\displaystyle {\begin{pmatrix}1&0\\k&1\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}x-a\\y-b\end{pmatrix}}}$ + ${\displaystyle {\begin{pmatrix}a\\b\end{pmatrix}}}$

Rumus sederhana

Keterangan	Posisi	Hasil
Translasi
penggeseran (a,b)	${\displaystyle (x,y)}$	${\displaystyle (x+a,y+b)}$
Refleksi
sumbu x [0°]	${\displaystyle (x,y)}$	${\displaystyle (x,-y)}$
sumbu y [90°]	${\displaystyle (x,y)}$	${\displaystyle (-x,y)}$
y=x [45°]	${\displaystyle (x,y)}$	${\displaystyle (y,x)}$
y=-x [135°]	${\displaystyle (x,y)}$	${\displaystyle (-y,-x)}$
pusat (0,0) [0° dan 90°]	${\displaystyle (x,y)}$	${\displaystyle (-x,-y)}$
pusat (a,b) [0° dan 90°]	${\displaystyle (x,y)}$	${\displaystyle (2a-x,2b-y)}$
pusat (a,0) [0° dan 90°]	${\displaystyle (x,y)}$	${\displaystyle (2a-x,y)}$
pusat (0,b) [0° dan 90°]	${\displaystyle (x,y)}$	${\displaystyle (x,2b-y)}$
Rotasi
berpusat (0,0)
90°	${\displaystyle (x,y)}$	${\displaystyle (-y,x)}$
-90°	${\displaystyle (x,y)}$	${\displaystyle (y,-x)}$
180°	${\displaystyle (x,y)}$	${\displaystyle (-x,-y)}$
berpusat (a,b)
90°	${\displaystyle (x,y)}$	${\displaystyle (-y+a+b,x-a+b)}$
-90°	${\displaystyle (x,y)}$	${\displaystyle (y-a+b,-x+a+b)}$
180°	${\displaystyle (x,y)}$	${\displaystyle (-x+2a,-y+2b)}$
berpusat (0,0)
Dilatasi
skala k	${\displaystyle (x,y)}$	${\displaystyle (k\cdot x,k\cdot y)}$
Stretching
sumbu x dan skala k	${\displaystyle (x,y)}$	${\displaystyle (k\cdot x,y)}$
sumbu y dan skala k	${\displaystyle (x,y)}$	${\displaystyle (x,k\cdot y)}$
Shearing
sumbu x dan skala k	${\displaystyle (x,y)}$	${\displaystyle (k\cdot x+y,y)}$
sumbu y dan skala k	${\displaystyle (x,y)}$	${\displaystyle (x,x+k\cdot y)}$
berpusat (a,b)
Dilatasi
skala k	${\displaystyle (x,y)}$	${\displaystyle (k\cdot x+(1-k)a,k\cdot y+(1-k)b)}$
Stretching
sumbu x dan skala k	${\displaystyle (x,y)}$	${\displaystyle (k\cdot x+(1-k)a,y)}$
sumbu y dan skala k	${\displaystyle (x,y)}$	${\displaystyle (x,k\cdot y+(1-k)b)}$
Shearing
sumbu x dan skala k	${\displaystyle (x,y)}$	${\displaystyle (x+k\cdot (y-b)),y)}$
sumbu y dan skala k	${\displaystyle (x,y)}$	${\displaystyle (x,y+k\cdot (x-a))}$