Roboguru

Diketahui: ,  Tentukan: b. Panjang proyeksi vektor dari  ke

Pertanyaan

Diketahui: a with rightwards harpoon with barb upwards on top equals 2 i with rightwards harpoon with barb upwards on top plus 2 j with rightwards harpoon with barb upwards on top minus k with rightwards harpoon with barb upwards on topb with rightwards harpoon with barb upwards on top equals 6 i with rightwards harpoon with barb upwards on top minus 3 j with rightwards harpoon with barb upwards on top plus 2 k with rightwards harpoon with barb upwards on top

Tentukan:

b. Panjang proyeksi vektor dari b ke a

Pembahasan Soal:

Diperoleh hasil panjang ooyeksi yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar b subscript a close vertical bar end cell equals cell fraction numerator open vertical bar a times b close vertical bar over denominator open vertical bar a close vertical bar end fraction end cell row blank equals cell fraction numerator open vertical bar 2 open parentheses 6 close parentheses plus 2 open parentheses negative 3 close parentheses plus open parentheses open parentheses negative 1 close parentheses 2 close parentheses close vertical bar over denominator square root of 2 squared plus 2 squared plus open parentheses negative 1 close parentheses squared end root end fraction end cell row blank equals cell fraction numerator open vertical bar 12 plus open parentheses negative 6 close parentheses plus open parentheses negative 2 close parentheses close vertical bar over denominator square root of 4 plus 4 plus 1 end root end fraction end cell row blank equals cell fraction numerator open vertical bar 12 minus 6 minus 2 close vertical bar over denominator square root of 9 end fraction end cell row blank equals cell 4 over 3 end cell end table

Maka, panjang proyeksi vektor dari a ke b adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 4 over 3 end cell end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Ayu

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Terakhir diupdate 05 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Sebuah vektor  diproyeksikan pada vektor . Panjang proyeksi vektor yang dihasilkan adalah ...

Pembahasan Soal:

Diketahui vektor A with rightwards arrow on top equals open parentheses negative 2 comma 4 close parentheses diproyeksi pada B with rightwards arrow on top equals open parentheses 3 comma negative 1 close parentheses.

Panjang proyeksi vektor A with rightwards arrow on top equals open parentheses negative 2 comma 4 close parentheses pada B with rightwards arrow on top equals open parentheses 3 comma negative 1 close parentheses:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar p close vertical bar end cell equals cell fraction numerator open vertical bar A with rightwards arrow on top times B with rightwards arrow on top close vertical bar over denominator open vertical bar B with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator open vertical bar open parentheses table row cell negative 2 end cell row 4 end table close parentheses times open parentheses table row 3 row cell negative 1 end cell end table close parentheses close vertical bar over denominator square root of 3 squared plus open parentheses negative 1 close parentheses squared end root end fraction end cell row blank equals cell fraction numerator open vertical bar negative 2 open parentheses 3 close parentheses plus 4 open parentheses negative 1 close parentheses close vertical bar over denominator square root of 9 plus 1 end root end fraction end cell row blank equals cell fraction numerator open vertical bar negative 6 minus 4 close vertical bar over denominator square root of 10 end fraction end cell row blank equals cell fraction numerator open vertical bar negative 10 close vertical bar over denominator square root of 10 end fraction end cell row blank equals cell fraction numerator 10 over denominator square root of 10 end fraction end cell row blank equals cell fraction numerator 10 over denominator square root of 10 end fraction cross times fraction numerator square root of 10 over denominator square root of 10 end fraction end cell row blank equals cell fraction numerator 10 square root of 10 over denominator 10 end fraction end cell row blank equals cell square root of 10 end cell end table 

Jadi, panjang proyeksi vektor yang dihasilkan adalah square root of 10.

0

Roboguru

Diketahui  dan . Proyeksi skalar  pada  adalah 4 dan . Jika , maka nilai dari  adalah ....

Pembahasan Soal:

Karena begin mathsize 14px style open vertical bar italic b with rightwards arrow on top close vertical bar equals 9 end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar italic b with italic rightwards arrow on top close vertical bar end cell equals 9 row cell square root of 1 squared plus open parentheses negative sign 4 close parentheses squared plus italic y squared end root end cell equals 9 row cell square root of 1 plus 16 plus italic y squared end root end cell equals 9 row cell 1 plus 16 plus italic y squared end cell equals 81 row cell 17 plus italic y squared end cell equals 81 row cell italic y squared end cell equals 64 row italic y equals cell plus-or-minus 8 end cell end table end style 

Karena undefined, maka begin mathsize 14px style italic y equals 8 end style 

Oleh karena itu,

begin mathsize 14px style italic b with italic rightwards arrow on top equals open parentheses table row 1 row cell negative sign 4 end cell row italic y end table close parentheses equals open parentheses table row 1 row cell negative sign 4 end cell row 8 end table close parentheses end style 

Karena proyeksi skalar italic a with rightwards arrow on top pada begin mathsize 14px style italic b with italic rightwards arrow on top end style adalah 3, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator italic a with italic rightwards arrow on top ⋅ italic b with rightwards arrow on top over denominator open vertical bar italic b with rightwards arrow on top close vertical bar end fraction end cell equals 4 row cell fraction numerator open parentheses table row cell negative sign 8 end cell row italic x row 4 end table close parentheses ⋅ open parentheses table row 1 row cell negative sign 4 end cell row 8 end table close parentheses over denominator 9 end fraction end cell equals 4 row cell fraction numerator negative sign 8 ⋅ 1 plus italic x ⋅ open parentheses negative sign 4 close parentheses plus 4 ⋅ 8 over denominator 9 end fraction end cell equals 4 row cell fraction numerator negative sign 8 minus sign 4 italic x plus 32 over denominator 9 end fraction end cell equals 4 row cell fraction numerator negative sign 4 italic x plus 24 over denominator 9 end fraction end cell equals 4 row cell negative sign 4 italic x plus 24 end cell equals 36 row cell negative sign 4 italic x end cell equals 12 row italic x equals cell negative sign 3 end cell end table end style 

Oleh karena itu,

begin mathsize 14px style italic a with italic rightwards arrow on top equals open parentheses table row cell negative sign 8 end cell row italic x row 4 end table close parentheses equals open parentheses table row cell negative sign 8 end cell row cell negative sign 3 end cell row 4 end table close parentheses end style 

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar italic a with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative sign 8 close parentheses squared plus open parentheses negative sign 3 close parentheses squared plus 4 squared end root end cell row blank equals cell square root of 64 plus 9 plus 16 end root end cell row blank equals cell square root of 64 plus 9 plus 16 end root end cell row blank equals cell square root of 89 end cell end table 

Jadi, jawaban yang tepat adalah B.

0

Roboguru

Diketahui vektor  dan . Jika panjang proyeksi vektor  pada  adalah 2, nilai n = ...

Pembahasan Soal:

Proyeksi vektor begin mathsize 12px style p with rightwards harpoon with barb upwards on top end style pada begin mathsize 12px style q with rightwards harpoon with barb upwards on top end style adalah begin mathsize 12px style r with rightwards harpoon with barb upwards on top end style

Maka berlaku begin mathsize 12px style open vertical bar r with rightwards harpoon with barb upwards on top close vertical bar equals fraction numerator p with rightwards harpoon with barb upwards on top times q with rightwards harpoon with barb upwards on top over denominator open vertical bar q with rightwards harpoon with barb upwards on top close vertical bar end fraction end style

Diketahui:

  • begin mathsize 12px style p with rightwards harpoon with barb upwards on top equals i with rightwards harpoon with barb upwards on top minus j with rightwards harpoon with barb upwards on top plus 2 k with rightwards harpoon with barb upwards on top end style
  • begin mathsize 12px style q with rightwards harpoon with barb upwards on top equals 2 i with rightwards harpoon with barb upwards on top minus 2 j with rightwards harpoon with barb upwards on top plus n k with rightwards harpoon with barb upwards on top end style
  • begin mathsize 12px style open vertical bar r with rightwards harpoon with barb upwards on top close vertical bar equals 2 end style

Maka, nilai n dapat ditentukan sebagai berikut.

begin mathsize 12px style fraction numerator p with rightwards harpoon with barb upwards on top times q with rightwards harpoon with barb upwards on top over denominator open vertical bar q with rightwards harpoon with barb upwards on top close vertical bar end fraction equals open vertical bar r with rightwards harpoon with barb upwards on top close vertical bar  left right double arrow fraction numerator 2 plus 2 plus 2 n over denominator square root of 4 plus 4 plus n squared end root end fraction equals 2  left right double arrow fraction numerator 4 plus 2 n over denominator square root of 8 plus n squared end root end fraction equals 2  left right double arrow 4 plus 2 n equals 2 square root of 8 plus n squared end root  left right double arrow open parentheses 4 plus 2 n close parentheses squared equals 4 open parentheses 8 plus n squared close parentheses  left right double arrow 16 plus 16 n plus 4 n squared equals 32 plus 4 n squared  left right double arrow 16 n equals 16  left right double arrow n equals 1 end style

Jadi, nilai n = 1

0

Roboguru

Diketahui A(4, 0, 0), B(0, -4, 0), dan C(0, 0, 4). Panjang vektor proyeksi  ke vektor  adalah...

Pembahasan Soal:

Jika titik Pbegin mathsize 12px style open parentheses a subscript 1 comma space a subscript 2 comma space a subscript 3 close parentheses end style dan Qbegin mathsize 12px style open parentheses b subscript 1 comma space b subscript 2 comma space b subscript 3 close parentheses end style , maka:

begin mathsize 12px style stack P Q with rightwards arrow on top equals Q minus P 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 minus 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 end style

Misal: begin mathsize 12px style P with rightwards arrow on top end style = begin mathsize 12px style open parentheses a subscript 1 comma space a subscript 2 comma space a subscript 3 close parentheses end style dan size 12px Q with size 12px rightwards arrow on top = open parentheses size 12px b subscript size 12px 1 size 12px comma size 12px space size 12px b subscript size 12px 2 size 12px comma size 12px space size 12px b subscript size 12px 3 close parentheses, maka:

begin mathsize 12px style P with rightwards arrow on top end style . size 12px Q with size 12px rightwards arrow on top = open parentheses size 12px a subscript size 12px 1 size 12px b subscript size 12px 1 size 12px plus size 12px a subscript size 12px 2 size 12px b subscript size 12px 2 size 12px plus size 12px a subscript size 12px 3 size 12px b subscript size 12px 3 close parentheses

open vertical bar size 12px P with size 12px rightwards arrow on top close vertical bar size 12px equals square root of open parentheses size 12px a subscript size 12px 1 close parentheses to the power of size 12px 2 size 12px plus open parentheses size 12px a subscript size 12px 2 close parentheses to the power of size 12px 2 size 12px plus open parentheses size 12px a subscript size 12px 3 close parentheses to the power of size 12px 2 end root

 

Misal size 12px Z with size 12px rightwards arrow on top adalah Proyeksi skalar vektor begin mathsize 12px style P with rightwards arrow on top end style pada vektor size 12px Q with size 12px rightwards arrow on top, maka:

size 12px Z with size 12px rightwards arrow on top size 12px equals fraction numerator size 12px P with size 12px rightwards arrow on top size 12px. size 12px Q with size 12px rightwards arrow on top over denominator open vertical bar size 12px Q with size 12px rightwards arrow on top close vertical bar end fraction

 

Diketahui A(4, 0, 0), B(0, -4, 0), dan C(0, 0, 4). Panjang vektor proyeksi begin mathsize 12px style stack A C with rightwards arrow on top end style ke vektor stack size 12px A size 12px B with size 12px rightwards arrow on top, yaitu:

stack size 12px A size 12px C with size 12px rightwards arrow on top size 12px equals size 12px C size 12px minus size 12px A size 12px equals open parentheses table row size 12px 0 row size 12px 0 row size 12px 4 end table close parentheses size 12px minus open parentheses table row size 12px 4 row size 12px 0 row size 12px 0 end table close parentheses size 12px equals open parentheses table row cell size 12px minus size 12px 4 end cell row size 12px 0 row size 12px 4 end table close parentheses  stack size 12px A size 12px B with size 12px rightwards arrow on top size 12px equals size 12px B size 12px minus size 12px A size 12px equals open parentheses table row size 12px 0 row cell size 12px minus size 12px 4 end cell row size 12px 0 end table close parentheses size 12px minus open parentheses table row size 12px 4 row size 12px 0 row size 12px 0 end table close parentheses size 12px equals open parentheses table row cell size 12px minus size 12px 4 end cell row cell size 12px minus size 12px 4 end cell row size 12px 0 end table close parentheses

Misal undefined adalah proyeksi vektor begin mathsize 12px style stack A C with rightwards arrow on top end style ke undefined, maka:

size 12px Z with size 12px rightwards arrow on top size 12px equals fraction numerator stack size 12px A size 12px C with size 12px rightwards arrow on top size 12px. stack size 12px A size 12px B with size 12px rightwards arrow on top over denominator open vertical bar stack size 12px A size 12px B with size 12px rightwards arrow on top close vertical bar end fraction size 12px equals fraction numerator open parentheses table row cell size 12px minus size 12px 4 end cell row size 12px 0 row size 12px 4 end table close parentheses size 12px times open parentheses table row cell size 12px minus size 12px 4 end cell row cell size 12px minus size 12px 4 end cell row size 12px 4 end table close parentheses over denominator square root of open parentheses size 12px minus size 12px 4 close parentheses to the power of size 12px 2 size 12px plus open parentheses size 12px minus size 12px 4 close parentheses to the power of size 12px 2 size 12px plus size 12px 0 to the power of size 12px 2 end root end fraction  size 12px Z with size 12px rightwards arrow on top size 12px equals fraction numerator size 12px 16 size 12px plus size 12px 0 size 12px plus size 12px 0 over denominator square root of size 12px 32 end fraction  size 12px Z with size 12px rightwards arrow on top size 12px equals fraction numerator size 12px 16 over denominator size 12px 4 square root of size 12px 2 end fraction  size 12px Z with size 12px rightwards arrow on top size 12px equals fraction numerator size 12px 4 over denominator square root of size 12px 2 end fraction  size 12px Z with size 12px rightwards arrow on top size 12px equals size 12px 2 square root of size 12px 2

 

0

Roboguru

Diketahui vektor a=(−1,1,2),u=(−1,c,2),danx=(−3,0,1). L1​adalah luas segitiga siku-siku yang dibentuk oleh a dan proyeksi a pada x. L2​ adalah luas segitiga siku-siku yang dibentuk oleh u dan proyeksi...

Pembahasan Soal:

Ingat Panjang proyeksi a pada b

Panjangproyeksi=bab

Misalkan proyeksi vektor undefined pada undefined adalah begin mathsize 14px style b with rightwards arrow on top end style. Segitiga dengan luas L1 dapat digambarkan sebagai berikut

Perhatikan bahwa 

a==(1)2+12+226

Kemudian

b===xax(3)2+02+12(1)(3)+10+21105

Diperoleh nilai p

p====a2b2(6)2(105)262527

Diperoleh luas segitiga (L1)

L1=====21bp21105272520740514081140

Bandingkan dengan L2

L181140L2===81L281L2140

Selanjutnya proyeksi u pada x. Misalkan dengan v. Dengan cara yang sama seperti sebelumnya, didapatkan segitiga dengan luas L2 sebagai berikut

Perhatikan bahwa 

u==(1)2+c2+22c2+5

Kemudian

v===xux(3)2+02+12(1)(3)+c0+21105

Diperoleh nilai 

q====u2v2(c2+5)2(105)2c2+525c2+25

Sehingga

L214052140025414002244482c2=======21vq21105c2+25c2+25(kuadratkankeduaruas)c2+25c2+25(kalikan2keduaruas)2c2+5443

Dengan demikian, nilai dari 2c2=443.

Oleh karena itu, jawaban yang benar 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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