Roboguru

1. Diketahui vektor-vektor a=(2−1​),b=(6−2​),danc=(0−2​). Tentukan: a.  a∙b

Pertanyaan

1. Diketahui vektor-vektor begin mathsize 14px style a with rightwards arrow on top equals open parentheses table row 2 row cell negative 1 end cell end table close parentheses space comma space b with rightwards arrow on top equals open parentheses table row 6 row cell negative 2 end cell end table close parentheses space comma space dan space c with rightwards arrow on top equals open parentheses table row 0 row cell negative 2 end cell end table close parentheses. end style Tentukan:

a.  a with rightwards arrow on top bullet b with rightwards arrow on top  

Pembahasan Soal:

begin mathsize 14px style a with rightwards harpoon with barb upwards on top bullet b with rightwards arrow on top equals open parentheses stack a subscript i with rightwards arrow on top cross times b with rightwards arrow on top subscript i close parentheses plus open parentheses a with rightwards arrow on top subscript j cross times b with rightwards arrow on top subscript j close parentheses a with rightwards arrow on top bullet b with rightwards arrow on top equals open parentheses 2 cross times 6 close parentheses plus open parentheses negative 1 cross times negative 2 close parentheses a with rightwards arrow on top bullet b with rightwards arrow on top equals 12 plus 2 a with rightwards arrow on top bullet b with rightwards arrow on top equals 14 end style 

Dengan demikian, nilai a with rightwards arrow on top bullet b with rightwards arrow on top adalah 14.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Indah

Mahasiswa/Alumni Universitas Diponegoro

Terakhir diupdate 07 Oktober 2021

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Pertanyaan yang serupa

Diketahui . Besar sudut antara adanbadalah ....

Pembahasan Soal:

Dari soal tersebut, didapat bahwa begin mathsize 14px style a with rightwards arrow on top equals open parentheses table row 0 row cell negative 1 end cell row 1 end table close parentheses space d a n space b with rightwards arrow on top equals open parentheses table row 2 row cell negative 1 end cell row 2 end table close parentheses end style.

Oleh karena itu, kita dapatkan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell 0 times 2 plus open parentheses negative 1 close parentheses times open parentheses negative 1 close parentheses plus 1 times 2 end cell row blank equals cell 0 plus 1 plus 2 end cell row blank equals 3 end table end style

Kemudian, kita dapatkan pula bahwa

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 0 squared plus open parentheses negative 1 close parentheses squared plus 1 squared end root end cell row blank equals cell square root of 0 plus 1 plus 1 end root end cell row blank equals cell square root of 2 end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar b with rightwards arrow on top close vertical bar end cell equals cell square root of 2 squared plus open parentheses negative 1 close parentheses squared plus open parentheses negative 2 close parentheses squared end root end cell row blank equals cell square root of 4 plus 1 plus 4 end root end cell row blank equals cell square root of 9 end cell row blank equals 3 end table end style

Misalkan besar sudut antara begin mathsize 14px style a with rightwards arrow on top space d a n space b with rightwards arrow on top end style adalah α, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row cell cos space alpha end cell equals cell fraction numerator 3 over denominator square root of 2 times 3 end fraction end cell row cell cos space alpha end cell equals cell fraction numerator 1 over denominator square root of 2 end fraction end cell row cell cos space alpha end cell equals cell fraction numerator 1 over denominator square root of 2 end fraction times fraction numerator square root of 2 over denominator square root of 2 end fraction end cell row cell cos space alpha end cell equals cell 1 half square root of 2 end cell end table end style

Didapat bahwa α = 45°.

Jadi, jawaban yang tepat adalah B.

0

Roboguru

Diketahui vektor a=6xi+2xj​−8k, b=−4i+8j​+10k, dan c=−2i+3j​−5k. Jika vektor a tegak lurus b, maka vektor a−c=...

Pembahasan Soal:

Misalkan a with rightwards arrow on top equals open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell row cell a subscript 3 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 row cell b subscript 3 end cell end table close parentheses. Jika vektor a with rightwards arrow on top tegak lurus b with rightwards arrow on top, maka berlaku:

a with rightwards arrow on top times b with rightwards arrow on top equals 0 

Dimana:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell row cell a subscript 3 end cell end table close parentheses times open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell row cell b subscript 3 end cell end table close parentheses end cell row blank equals cell a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3 end cell end table 

Diketahui:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell 6 x i with rightwards arrow on top plus 2 x j with rightwards arrow on top minus 8 k with rightwards arrow on top end cell row cell b with rightwards arrow on top end cell equals cell negative 4 i with rightwards arrow on top plus 8 j with rightwards arrow on top plus 10 k with rightwards arrow on top end cell row cell c with rightwards arrow on top end cell equals cell negative 2 i with rightwards arrow on top plus 3 j with rightwards arrow on top minus 5 k with rightwards arrow on top end cell end table 

Tentukan nilai x vektor a with rightwards arrow on top menggunakan rumus dua vektor saling tegak lurus.

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals 0 row cell open parentheses table row cell 6 x end cell row cell 2 x end cell row cell negative 8 end cell end table close parentheses times open parentheses table row cell negative 4 end cell row 8 row 10 end table close parentheses end cell equals 0 row cell 6 x times open parentheses negative 4 close parentheses plus 2 x times 8 plus open parentheses negative 8 close parentheses times 10 end cell equals 0 row cell negative 24 x plus 16 x minus 80 end cell equals 0 row cell negative 8 x end cell equals 80 row x equals cell fraction numerator 80 over denominator negative 8 end fraction end cell row x equals cell negative 10 end cell end table 

Sehingga, vektor a with rightwards arrow on top adalah: 

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell open parentheses table row cell 6 x end cell row cell 2 x end cell row cell negative 8 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 6 times open parentheses negative 10 close parentheses end cell row cell 2 times open parentheses negative 10 close parentheses end cell row cell negative 8 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 60 end cell row cell negative 20 end cell row cell negative 8 end cell end table close parentheses end cell end table 

Maka, a with rightwards arrow on top minus c with rightwards arrow on top adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top minus c with rightwards arrow on top end cell equals cell open parentheses table row cell negative 60 end cell row cell negative 20 end cell row cell negative 8 end cell end table close parentheses minus open parentheses table row cell negative 2 end cell row 3 row cell negative 5 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 60 plus 2 end cell row cell negative 20 minus 3 end cell row cell negative 8 plus 5 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 58 end cell row cell negative 23 end cell row cell negative 3 end cell end table close parentheses end cell end table 

Sehingga, vektor a with rightwards arrow on top minus c with rightwards arrow on top equals negative 58 i with rightwards arrow on top minus 23 j with rightwards arrow on top minus 3 k with rightwards arrow on top.

Jadi, jawaban yang tepat adalah B.

0

Roboguru

Diketahuivektora=2i+j+3kdanb=−i+2j+2k.Sudutθadalahsudutantaravektoradanb.Nilaisinθ=....

Pembahasan Soal:

I n g a t space r u m u s space p e r k a l i a n space t i t i k space p a d a space v e k t o r space colon space  a space ⃗. b space ⃗ equals vertical line a space ⃗ space vertical line vertical line b space ⃗ space vertical line space space cos invisible function application A    a with rightwards arrow on top equals open parentheses table row 2 row 1 row 3 end table close parentheses space dan space b with rightwards arrow on top equals open parentheses table row cell negative 1 end cell row 2 row 2 end table close parentheses  open parentheses table row 2 row 1 row 3 end table close parentheses open parentheses table row cell negative 1 end cell row 2 row 2 end table close parentheses equals square root of 2 squared plus 1 squared plus 3 squared space end root space. space square root of open parentheses negative 1 close parentheses squared plus open parentheses 2 close parentheses squared plus open parentheses 2 close parentheses squared end root space cos invisible function application theta  minus 2 plus 2 plus 6 equals square root of left parenthesis 4 plus 1 plus 9 right parenthesis end root space space. square root of left parenthesis 1 plus 4 plus 4 right parenthesis end root space space cos invisible function application theta space  6 equals square root of 14 space space.3 space cos invisible function application theta  cos invisible function application theta equals fraction numerator 6 over denominator 3 square root of 14 end fraction equals fraction numerator 2 over denominator square root of 14 end fraction    P e r h a t i k a n space s e g i t i g a

M a k a space sin invisible function application theta equals fraction numerator square root of 10 over denominator square root of 14 end fraction equals 1 over 14 square root of 140 equals 2 over 14 square root of 35 equals fraction numerator square root of 35 over denominator 7 end fraction

0

Roboguru

Tentukan a⋅b jika diketahui: a=3i−4jdanb=3i+j

Pembahasan Soal:

Dengan memperhatikan konsep dot product dan sifat-sifatnya, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell a times b end cell equals cell open parentheses table row 3 row cell negative 4 end cell end table close parentheses times open parentheses table row 3 row 1 end table close parentheses end cell row blank equals cell open parentheses table row 9 row cell negative 4 end cell end table close parentheses end cell row blank equals cell 9 i minus 4 j end cell end table

Oleh karena itu, table attributes columnalign right center left columnspacing 0px end attributes row cell a times b end cell equals cell 9 i minus 4 j end cell end table.

0

Roboguru

Diketahui vektor a dan b membentuk sudut 120∘. Jika  dan , maka hasil dari b⋅(a−b) adalah ....

Pembahasan Soal:

Karena besar sudut yang terbentuk dari vektor undefined dan undefined adalah begin mathsize 14px style 120 degree end style, maka begin mathsize 14px style alpha equals 120 degree end style.

Kemudian, perhatikan hubungan berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell b with rightwards arrow on top times open parentheses a with rightwards arrow on top minus b with rightwards arrow on top close parentheses end cell equals cell b with rightwards arrow on top times a with rightwards arrow on top minus b with rightwards arrow on top times b with rightwards arrow on top end cell row blank equals cell a with rightwards arrow on top times b with rightwards arrow on top minus b with rightwards arrow on top times b with rightwards arrow on top end cell end table end style

Ingat bahwa begin mathsize 14px style a with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar cos alpha end style dan begin mathsize 14px style b with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar b with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar end style.

Oleh karena itu, kita dapatkan perhitungan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell b with rightwards arrow on top times open parentheses a with rightwards arrow on top minus b with rightwards arrow on top close parentheses end cell equals cell a with rightwards arrow on top times b with rightwards arrow on top minus b with rightwards arrow on top times b with rightwards arrow on top end cell row blank equals cell open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar cos space alpha minus open vertical bar b with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar end cell row blank equals cell 5 times 2 times cos space 120 degree minus 2 times 2 end cell row blank equals cell 5 times 2 times open parentheses negative 1 half close parentheses minus 2 times 2 end cell row blank equals cell negative 5 minus 4 end cell row blank equals cell negative 9 end cell end table end style

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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