Soal-Soal Matematika/Vektor

Vektor

Posisi vektor

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

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

Panjang vektor

Berada di $R^{2}$

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

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

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

Berada di $R^{3}$

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

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

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

Jumlah dan selisih kedua vektor

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

Vektor satuan

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

Operasi aljabar pada vektor

• Penjumlahan dan pengurangan

terdiri dari 2 aturan jenis yaitu aturan segitiga dan jajar genjang

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

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

• Perkalian

1. skalar dengan vektor

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

1. titik dua vektor

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

1. titik dua vektor dengan membentuk sudut

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

1. silang dua vektor

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

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

1. silang dua vektor dengan membentuk sudut

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

Sifat operasi aljabar pada vektor

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

Hubungan vektor dengan vektor lain

• Perkalian titik

Saling tegak lurus

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

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

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

Sejajar

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

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

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

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

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

• Perkalian silang

Saling tegak lurus

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

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

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

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

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

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

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

Sejajar

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

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

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

Sudut dua vektor

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

Panjang proyeksi dan proyeksi vektor

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

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

Metode

segitiga

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

jajar genjang

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

Perbandingan

Aturan jajar genjang

Posisi vektor

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

Berada di $R^{2}$

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

Berada di $R^{3}$

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

Satu garis

• Perbandingan posisi dalam adalah m:n

Posisi vektor

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

Berada di $R^{2}$

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

Berada di $R^{3}$

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

• Perbandingan posisi luar adalah m:-n

Posisi vektor

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

Berada di $R^{2}$

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

Berada di $R^{3}$

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

contoh

1. Titik a -3i-2j+4k dan b 6i+6j+k. tentukan:

• ${\vec {a}}+{\vec {b}}$
• ${\vec {a}}-{\vec {b}}$
• ${\vec {a}}\cdot {\vec {b}}$
• ${\vec {a}}\times {\vec {b}}$
• panjang vektor
• vektor satuan pada vektor b
• panjang proyeksi
• proyeksi vektor

Jawaban

$${\begin{aligned}*{\vec {c}}&={\vec {a}}+{\vec {b}}\\&=(-3,-2,4)+(6,6,1)\\&=(-3+6),(-2+6),(4+1)\\&=3,4,5\\*{\vec {c}}&={\vec {a}}-{\vec {b}}\\&=(-3,-2,4)-(6,6,1)\\&=(-3-6),(-2-6),(4-1)\\&=-9,-8,3\\*{\vec {a}}\cdot {\vec {b}}\\&=(-3,-2,4)\cdot (6,6,1)\\&=(-3\cdot 6)+(-2\cdot 6)+(4\cdot 1)\\&=-9-8+3\\&=-14\\*{\vec {a}}\times {\vec {b}}\\{\begin{array}{rrr|rr}i&j&k&i&j\\-3&-2&4&-3&-2\\6&6&1&6&6\\-&-&-+&+&+\\\end{array}}\\&=(-2\cdot 1)i+(4\cdot 6)j+(-3\cdot 6)k-(-3\cdot 1)j-(4\cdot 6)i-(-2\cdot 6)k\\&=-2i+24j-18k+3j-24i+12k\\&=-26i+27j-6k\\*|{\vec {c}}|&={\sqrt {(6-(-3))^{2}+(6-(-2))^{2}+(1-4)^{2}}}\\&={\sqrt {9^{2}+8^{2}+(-3)^{2}}}\\&={\sqrt {81+64+9}}\\&={\sqrt {81+64+9}}\\&={\sqrt {145}}\\*{\hat {b}}&={\frac {\vec {b}}{|{\vec {b}}|}}\\&={\frac {6,6,1}{\sqrt {6^{2}+6^{2}+1^{2}}}}\\&={\frac {6,6,1}{\sqrt {36+36+1}}}\\&={\frac {6,6,1}{\sqrt {73}}}\\*|{\vec {c}}|&={\frac {{\vec {a}}\cdot {\vec {b}}}{|{\vec {b}}|}}\\&={\frac {14}{\sqrt {73}}}\\*{\vec {c}}&={\frac {{\vec {a}}\cdot {\vec {b}}}{|{\vec {b}}|^{2}}}\cdot {\vec {b}}\\&={\frac {14}{({\sqrt {73}})^{2}}}(6,6,1)\\&={\frac {14}{73}}(6,6,1)\\\end{aligned}}$$