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Diketahui vektor a=2i+j dan b=3i−4j  Tentukan : 4. Panjang vektor a−b

Pertanyaan

Diketahui vektor begin mathsize 14px style straight a equals 2 straight i plus straight j space end style dan begin mathsize 14px style straight b equals 3 straight i minus 4 straight j end style 

Tentukan :

4. Panjang vektor begin mathsize 14px style straight a minus straight b end style 

Pembahasan Soal:

Kita tentukan dahulu vektor begin mathsize 14px style straight a minus straight b end style
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell top enclose straight a minus top enclose straight b end cell equals cell open parentheses 2 straight i plus straight j close parentheses minus open parentheses 3 straight i minus 4 straight j close parentheses end cell row blank equals cell open parentheses 2 minus 3 close parentheses straight i plus open parentheses 1 minus open parentheses negative 4 close parentheses close parentheses straight j end cell row blank equals cell negative 1 straight i plus open parentheses negative 3 close parentheses straight j end cell row blank equals cell open parentheses negative 1 comma space minus 3 close parentheses end cell end table end style 
 

Panjang vektor :
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open vertical bar top enclose straight a minus top enclose straight b close vertical bar end cell equals cell square root of straight x squared plus straight y squared end root end cell row blank equals cell square root of open parentheses negative 1 close parentheses squared plus open parentheses negative 3 close parentheses squared end root end cell row blank equals cell square root of 1 plus 9 end root end cell row blank equals cell square root of 10 end cell end table end style 


Jadi, Panjang vektor begin mathsize 14px style straight a minus straight b end style  adalah begin mathsize 14px style square root of 10 end style 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Terakhir diupdate 05 Oktober 2021

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Pertanyaan yang serupa

Sebuah vektor x dengan panjang 22​ membuat sudut lancip dengan y​=[3,1]. Bila vektor  diproyeksikan ke vektor y​ panjang proyeksinya sama dengan 54​10​. Vektor  tersebut adalah ...

Pembahasan Soal:

Panjang vektor a with rightwards arrow on top equals open parentheses table row x row y end table close parentheses, yaitu open vertical bar a with rightwards arrow on top close vertical bar equals square root of x squared plus y squared end root

Jika vektor a with rightwards arrow on top equals open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell end table close parentheses dan b with rightwards arrow on top equals open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell end table close parentheses, maka

a with rightwards arrow on top times b with rightwards arrow on top equals a subscript 1 b subscript 1 plus a subscript 2 b subscript 2

Panjang proyeksi vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top dapat ditentukan menggunakan rumus berikut.

open vertical bar c with rightwards arrow on top close vertical bar equals fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction

Berdasarkan konsep di atas dapat ditentukan hubungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar c with rightwards arrow on top close vertical bar end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator open vertical bar y close vertical bar end fraction end cell row cell 4 over 5 square root of 10 end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator square root of 3 squared plus 1 squared end root end fraction end cell row cell 4 over 5 square root of 10 end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator square root of 10 end fraction end cell row cell x with rightwards arrow on top times y with rightwards arrow on top end cell equals cell fraction numerator 4 square root of 10 times square root of 10 over denominator 5 end fraction end cell row cell x with rightwards arrow on top times y with rightwards arrow on top end cell equals 8 end table

Misal vektor x with rightwards arrow on top equals open square brackets a comma space b close square brackets dapat ditentukan persamaan-persamaan berikut.

Persamaan 1:

table attributes columnalign right center left columnspacing 0px end attributes row cell x with rightwards arrow on top times y with rightwards arrow on top end cell equals 8 row cell open parentheses table row a row b end table close parentheses open parentheses table row 3 row 1 end table close parentheses end cell equals 8 row cell 3 a plus b end cell equals 8 row b equals cell 8 minus 3 a end cell end table

Persamaan 2:

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of a squared plus b squared end root end cell equals cell open vertical bar x with rightwards arrow on top close vertical bar end cell row cell square root of a squared plus b squared end root end cell equals cell 2 square root of 2 end cell row cell a squared plus b squared end cell equals 8 end table

Substitusi persamaan 1 ke persamaan 2:

table attributes columnalign right center left columnspacing 0px end attributes row cell a squared plus b squared end cell equals 8 row cell a squared plus open parentheses 8 minus 3 a close parentheses squared end cell equals 8 row cell a squared plus 64 minus 48 a plus 9 a squared end cell equals 8 row cell 10 a squared minus 48 a plus 56 end cell equals 0 row cell 5 a squared minus 24 a plus 28 end cell equals 0 row cell open parentheses a minus 2 close parentheses open parentheses 5 a minus 14 close parentheses end cell equals 0 end table

a equals 2 atau a equals 14 over 5

Untuk a equals 2 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row b equals cell 8 minus 3 a end cell row blank equals cell 8 minus 3 times 2 end cell row blank equals 2 end table

Untuk a equals 14 over 5 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row b equals cell 8 minus 3 a end cell row blank equals cell 8 minus 3 times 14 over 5 end cell row blank equals cell 40 over 5 minus 42 over 5 end cell row blank equals cell negative 2 over 5 end cell end table

Vektor x with rightwards arrow on top equals open square brackets 2 comma space 2 close square brackets space text atau end text space open square brackets 14 over 5 comma space minus 2 over 5 close square brackets

Oleh karena itu, jawaban yang tepat adalah A.

0

Roboguru

Tentukan komponen-komponen dari vektor-vektor berikut.

Pembahasan Soal:

Perhatikan 

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell p with rightwards harpoon with barb upwards on top end cell equals cell open parentheses table row 3 row 2 end table close parentheses end cell row cell q with rightwards harpoon with barb upwards on top end cell equals cell open parentheses table row 3 row 4 end table close parentheses end cell row cell r with rightwards harpoon with barb upwards on top end cell equals cell open parentheses table row 5 row 0 end table close parentheses end cell row cell VB with rightwards harpoon with barb upwards on top end cell equals cell open parentheses table row 3 row 4 end table close parentheses end cell row cell DR with rightwards harpoon with barb upwards on top end cell equals cell open parentheses table row 3 row 2 end table close parentheses end cell end table end style 

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Roboguru

Tentukan besar vektor AB jikaA(−2,3) dan B(1,−4).

Pembahasan Soal:

Jika A open parentheses italic x subscript italic 1 italic comma italic y subscript 1 close parentheses dan B open parentheses italic x subscript italic 2 italic comma italic y subscript italic 2 close parentheses, maka panjang AB adalah

open vertical bar AB close vertical bar equals square root of open parentheses italic y subscript 2 minus sign italic y subscript 1 close parentheses squared plus open parentheses italic x subscript 2 minus sign x subscript 1 close parentheses squared end root  

Besar vektor AB jikabegin mathsize 14px style space A left parenthesis minus sign 2 comma 3 right parenthesis end style dan begin mathsize 14px style B left parenthesis 1 comma minus sign 4 right parenthesis end style, maka panjang AB adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar AB close vertical bar end cell equals cell square root of open parentheses italic y subscript 2 minus sign italic y subscript 1 close parentheses squared plus open parentheses italic x subscript 2 minus sign x subscript 1 close parentheses squared end root end cell row blank equals cell square root of open parentheses negative sign 4 minus sign 3 close parentheses squared plus open parentheses 1 minus sign open parentheses negative sign 2 close parentheses close parentheses squared end root end cell row blank equals cell square root of open parentheses negative sign 7 close parentheses squared plus open parentheses 3 close parentheses squared end root end cell row blank blank cell square root of 49 plus 9 end root end cell row blank equals cell square root of 58 end cell end table

Jadi,  besar vektor AB jikabegin mathsize 14px style space A left parenthesis minus sign 2 comma 3 right parenthesis end style dan begin mathsize 14px style B left parenthesis 1 comma minus sign 4 right parenthesis end style adalah square root of 58.

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Roboguru

Diketahui koordinat titik T(−4,2),U(−1,1),danV(8,−2), jika TV=kUT, nilai k = ...

Pembahasan Soal:

Diketahui:

T left parenthesis negative 4 comma 2 right parenthesis comma U left parenthesis negative 1 comma 1 right parenthesis comma space dan space V left parenthesis 8 comma negative 2 right parenthesis

stack T V with rightwards arrow on top equals k space stack U T with rightwards arrow on top

Ditanya: k?

Jawab:

table attributes columnalign right center left columnspacing 0px end attributes row cell stack T V with rightwards arrow on top end cell equals cell k space stack U T with rightwards arrow on top end cell row cell open parentheses table row 8 row cell negative 2 end cell end table close parentheses minus open parentheses table row cell negative 4 end cell row 2 end table close parentheses end cell equals cell k open parentheses open parentheses table row cell negative 4 end cell row 2 end table close parentheses minus open parentheses table row cell negative 1 end cell row 1 end table close parentheses close parentheses end cell row cell open parentheses table row 12 row cell negative 4 end cell end table close parentheses end cell equals cell k open parentheses table row cell negative 3 end cell row 1 end table close parentheses end cell row blank blank blank row 12 equals cell negative 3 k end cell row k equals cell negative 4 end cell end table

Oleh karena itu, jawaban yang benar adalah D.

 

0

Roboguru

Jika vektor a=(−36​) maka ∣∣​a∣∣​ adalah... .

Pembahasan Soal:

Misalkan vektor a with rightwards arrow on top equals open parentheses a subscript 1 comma space a subscript 2 close parentheses maka panjang vektor pada dua dimensi digunakan rumus berikut:

open vertical bar a with rightwards arrow on top close vertical bar equals square root of open parentheses a subscript 1 close parentheses squared plus open parentheses a subscript 2 close parentheses squared end root 

Diketahui begin mathsize 14px style a with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 6 end table close parentheses end style maka begin mathsize 14px style open vertical bar a with rightwards arrow on top close vertical bar end style sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 3 close parentheses squared plus open parentheses 6 close parentheses squared end root end cell row blank equals cell square root of 9 plus 36 end root end cell row blank equals cell square root of 45 end cell row blank equals cell square root of 9 cross times 5 end root end cell row blank equals cell 3 square root of 5 end cell end table end style

Dengan demikian, panjang vektor a with rightwards arrow on top adalah 3 square root of 5.

Oleh karena itu, jawaban yang benar adalah A.

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