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Diketahui vektor . a. Hitunglah vektor satuan dari ! b. Tunjukkan bahwa panjang vektor satuan dari  adalah 1!

Pertanyaan

Diketahui vektor begin mathsize 14px style m with rightwards arrow on top equals open parentheses 1 comma space minus 5 close parentheses end style.

a. Hitunglah vektor satuan dari begin mathsize 14px style m with rightwards arrow on top end style!

b. Tunjukkan bahwa panjang vektor satuan dari begin mathsize 14px style m with rightwards arrow on top end style adalah 1!

Pembahasan Soal:

a. Panjang vektor begin mathsize 14px style m with rightwards arrow on top end style dapat ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar m with rightwards arrow on top close vertical bar end cell equals cell square root of 1 squared plus open parentheses negative 5 close parentheses squared end root end cell row blank equals cell square root of 1 plus 25 end root end cell row blank equals cell square root of 26 end cell end table end style 

Misalkan vektor satuan dari begin mathsize 14px style m with rightwards arrow on top end style adalah vektor begin mathsize 14px style u with rightwards arrow on top end style. Dengan menerapkan rumus vektor satuan, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell u with rightwards arrow on top end cell equals cell fraction numerator m with rightwards arrow on top over denominator open vertical bar m with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of 26 end fraction open parentheses table row 1 row cell negative 5 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator 1 over denominator square root of 26 end fraction end cell row cell fraction numerator negative 5 over denominator square root of 26 end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 over 26 square root of 26 end cell row cell negative 5 over 26 square root of 26 end cell end table close parentheses end cell end table end style 

Jadi, vektor satuan dari begin mathsize 14px style m with rightwards arrow on top end style adalah begin mathsize 14px style u with rightwards arrow on top equals open parentheses table row cell 1 over 26 square root of 26 end cell row cell negative 5 over 26 square root of 26 end cell end table close parentheses end style.


b. Panjang vektor begin mathsize 14px style u with rightwards arrow on top end style dapat ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar u with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses fraction numerator 1 over denominator square root of 26 end fraction close parentheses squared plus open parentheses fraction numerator negative 5 over denominator square root of 26 end fraction close parentheses squared end root end cell row blank equals cell square root of 1 over 26 plus 25 over 26 end root end cell row blank equals cell square root of 26 over 26 end root end cell row blank equals cell square root of 1 end cell row blank equals 1 end table end style 

Jadi, terbukti bahwa panjang vektor satuan dari begin mathsize 14px style m with rightwards arrow on top end style adalah 1 satuan.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Terakhir diupdate 29 Maret 2021

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Pertanyaan yang serupa

Hitunglah Vektor satuan dari Vektor

Pembahasan Soal:

Vektor satuan : table attributes columnalign right center left columnspacing 0px end attributes row cell u with rightwards arrow on top end cell equals cell fraction numerator left parenthesis negative 6 comma 8 right parenthesis over denominator square root of left parenthesis negative 6 right parenthesis squared plus 8 to the power of 2 end exponent end root end fraction end cell row blank equals cell fraction numerator left parenthesis negative 6 comma 8 right parenthesis over denominator square root of 36 plus 64 end root end fraction end cell row blank equals cell fraction numerator left parenthesis negative 6 comma 8 right parenthesis over denominator square root of 100 end fraction end cell row blank equals cell fraction numerator left parenthesis negative 6 comma 8 right parenthesis over denominator 10 end fraction end cell row blank equals cell open parentheses fraction numerator negative 3 over denominator 5 end fraction comma 4 over 5 close parentheses end cell end table

Jadi, vektor satuan u with rightwards arrow on top equals open parentheses fraction numerator negative 3 over denominator 5 end fraction comma 4 over 5 close parentheses.

0

Roboguru

Gambarkan vektor berikut!

Pembahasan Soal:

Vektor dibuat mulai dari titik undefined ke titik begin mathsize 14px style open parentheses negative 4 comma space 5 close parentheses end style.
Berikut ini adalah gambar vektor dari soal tersebut.

0

Roboguru

Diketahui vektor-vektor: Tentukan vektor satuan dari  !

Pembahasan Soal:

Ingat bahwa vektor satuan adalah vektor yang panjangnya sama dengan satu. Berdasarkan soal diketahui straight a with rightwards arrow on top equals open parentheses table row 6 row 0 end table close parentheses comma space straight b with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row cell negative 4 end cell end table close parentheses comma space straight c with rightwards arrow on top equals open parentheses table row 5 row 2 end table close parentheses comma space dan space straight d with rightwards arrow on top equals open parentheses table row 4 row cell negative 1 end cell end table close parentheses, sehingga panjang vektor straight a with rightwards arrow on top plus straight b with rightwards arrow on top minus 3 straight c with rightwards arrow on top:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight a with rightwards arrow on top plus straight b with rightwards arrow on top minus 3 straight c with rightwards arrow on top end cell equals cell open parentheses table row 6 row 0 end table close parentheses plus open parentheses table row cell negative 3 end cell row cell negative 4 end cell end table close parentheses minus 3 open parentheses table row 5 row 2 end table close parentheses end cell row blank equals cell open parentheses table row 6 row 0 end table close parentheses plus open parentheses table row cell negative 3 end cell row cell negative 4 end cell end table close parentheses minus open parentheses table row 15 row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 12 end cell row cell negative 10 end cell end table close parentheses end cell row cell open vertical bar straight a plus straight b minus 3 straight c close vertical bar end cell equals cell square root of left parenthesis negative 12 right parenthesis squared plus left parenthesis negative 10 right parenthesis to the power of 2 end exponent end root end cell row blank equals cell square root of 144 plus 100 end root end cell row blank equals cell square root of 244 end cell row blank equals cell 2 square root of 61 end cell end table


Jadi, vektor satuan straight a with rightwards arrow on top plus straight b with rightwards arrow on top minus 3 straight c with rightwards arrow on top:

table attributes columnalign right center left columnspacing 0px end attributes row cell vektor space satuan space straight a with rightwards arrow on top plus straight b with rightwards arrow on top minus 3 straight c with rightwards arrow on top end cell equals cell fraction numerator 1 over denominator open vertical bar straight a plus straight b minus 3 straight c close vertical bar end fraction times left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top minus 3 straight c with rightwards arrow on top right parenthesis end cell row blank equals cell fraction numerator 1 over denominator 2 square root of 61 end fraction open parentheses table row cell negative 12 end cell row cell negative 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator negative 6 over denominator square root of 61 end fraction end cell row cell fraction numerator negative 5 over denominator square root of 61 end fraction end cell end table close parentheses end cell end table

0

Roboguru

Diketahui , , . Tentukan panjang vector ,  dan  serta vector satuan yang searah dengan vector

Pembahasan Soal:

Vektor undefined  

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack A B with rightwards arrow on top end cell equals cell straight B minus straight A end cell row blank equals cell open parentheses table row cell negative 3 end cell row cell negative 4 end cell end table close parentheses minus open parentheses table row 6 row 0 end table close parentheses end cell row cell stack A B with rightwards arrow on top end cell equals cell open parentheses table row cell negative 9 end cell row cell negative 4 end cell end table close parentheses end cell end table end style 

Panjang vektor begin mathsize 14px style stack A B with rightwards arrow on top end style adalah 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack A B with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 9 close parentheses squared plus open parentheses negative 4 close parentheses squared end root end cell row blank equals cell square root of 81 plus 16 end root end cell row cell open vertical bar stack A B with rightwards arrow on top close vertical bar end cell equals cell square root of 97 end cell end table end style 

 

Vektor begin mathsize 14px style stack B C with rightwards arrow on top end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack B C with rightwards arrow on top end cell equals cell straight C minus straight B end cell row blank equals cell open parentheses table row 5 row 2 end table close parentheses minus open parentheses table row cell negative 3 end cell row cell negative 4 end cell end table close parentheses end cell row cell stack B C with rightwards arrow on top end cell equals cell open parentheses table row 8 row 6 end table close parentheses end cell end table end style 

Panjang vektor undefined adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack B C with rightwards arrow on top close vertical bar end cell equals cell square root of 8 squared plus 6 squared end root end cell row blank equals cell square root of 64 plus 36 end root end cell row blank equals cell square root of 100 end cell row cell open vertical bar stack B C with rightwards arrow on top close vertical bar end cell equals 10 end table end style 

Vektor begin mathsize 14px style stack A C with rightwards arrow on top end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack A C with rightwards arrow on top end cell equals cell straight C minus straight A end cell row blank equals cell open parentheses table row 5 row 2 end table close parentheses minus open parentheses table row 6 row 0 end table close parentheses end cell row cell stack A C with rightwards arrow on top end cell equals cell open parentheses table row cell negative 1 end cell row 2 end table close parentheses end cell end table end style 

Panjang vektor undefined adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack A C with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 1 close parentheses squared plus 2 squared end root end cell row blank equals cell square root of 1 plus 4 end root end cell row cell open vertical bar stack A C with rightwards arrow on top close vertical bar end cell equals cell square root of 5 end cell end table end style 

Vektor satuan yang searah dengan undefined adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack B C with rightwards arrow on top end cell equals cell fraction numerator stack B C with rightwards arrow on top over denominator open vertical bar stack B C with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell 1 over 10 open parentheses negative 3 comma space minus 4 close parentheses end cell row cell stack B C with rightwards arrow on top end cell equals cell open parentheses fraction numerator negative 3 over denominator 10 end fraction comma space fraction numerator negative 4 over denominator 10 end fraction close parentheses end cell end table end style 

Dengan demikian panjang vektor begin mathsize 14px style stack A B with rightwards arrow on top comma space stack B C with rightwards arrow on top end style dan undefined serta vektor satuan yang searah dengan vektor undefined adalah begin mathsize 14px style square root of 97 comma space 10 end style dan begin mathsize 14px style square root of 5 end style serta begin mathsize 14px style open parentheses fraction numerator negative 3 over denominator 10 end fraction comma space fraction numerator negative 4 over denominator 10 end fraction close parentheses end style.

0

Roboguru

Diketahui vector  maka vector satuan a adalah ...

Pembahasan Soal:

Langkah pertama untuk menyelesaikan soal di atas adalah dengan menentukan panjang vektor a.

Seperti yang kamu ketahui, kuadrat panjang vektor a sama dengan jumlah kuadrat komponen-komponennya.

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight a close vertical bar squared end cell equals cell 4 squared plus open parentheses negative 3 close parentheses squared end cell row blank equals cell 16 plus 9 end cell row blank equals 25 row blank equals cell square root of 25 end cell row blank equals cell plus-or-minus 5 end cell end table

Karena panjang tidak pernah negatif, maka nilai a = 5

Langkah kedua adalah menentukan vektor satuan yang searah dengan vektor a, yaitu fraction numerator a over denominator open vertical bar a close vertical bar end fraction.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator straight a over denominator open vertical bar straight a close vertical bar end fraction end cell equals cell fraction numerator 4 straight i minus 3 straight j over denominator 5 end fraction end cell row blank equals cell 4 over 5 straight i minus 3 over 5 straight j end cell end table

Jadi, dapat disimpulkan bahwa vektor satuan yang mempunyai arah sama dengan vektor straight a equals 4 straight i minus 3 straight j adalah vektor adalah 4 over 5 straight i space minus 3 over 5 straight j.

0

Roboguru

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