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Diketahui  dan . Tentukan panjang proyeksi dan vektor proyeksi a terhadap vektor b.

Pertanyaan

Diketahui straight a equals 2 straight i plus 2 straight j minus straight k dan straight b equals 6 straight i minus 3 straight j plus 2 straight k. Tentukan panjang proyeksi dan vektor proyeksi a terhadap vektor b.

Pembahasan Soal:

Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah

open vertical bar x with rightwards arrow on top close vertical bar equals open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar space dan space x with rightwards arrow on top equals fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction times v with rightwards arrow on top

maka, panjang proyeksi a terhadap vektor b adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell open vertical bar 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 close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses over denominator square root of 6 squared plus left parenthesis negative 3 right parenthesis squared plus 2 to the power of 2 end exponent end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 12 minus 6 minus 2 over denominator square root of 36 plus 9 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 4 over denominator square root of 49 end fraction close vertical bar end cell row blank equals cell open vertical bar 4 over 7 close vertical bar end cell row blank equals cell 4 over 7 end cell end table

dan proyeksi vektor a terhadap vektor b adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell x with rightwards arrow on top end cell equals cell 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 squared end fraction times b with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses over denominator open square brackets square root of 6 squared plus left parenthesis negative 3 right parenthesis squared plus 2 squared end root close square brackets squared end fraction times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell 4 over 49 open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell open parentheses table row cell 24 over 49 end cell row cell negative 12 over 49 end cell row cell 8 over 49 end cell end table close parentheses end cell end table

Oleh karena itu, panjang proyeksi dan vektor preyeksi a terhadap b berturut-turut adalah 4 over 7 space dan space 24 over 49 straight i minus 12 over 49 straight j plus 8 over 49 straight k.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Kurnia

Mahasiswa/Alumni Universitas Jember

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui  dan . O adalah titik pangkal. Tentukan panjang proyeksi dan vektor proyeksi p ke q.

Pembahasan Soal:

Diketahui P left parenthesis 2 comma space 4 comma space 3 right parenthesis dan Q left parenthesis 1 comma space minus 5 comma space 2 right parenthesis, maka

straight p equals 2 straight i plus 4 straight j plus 3 straight k space dan space straight q equals straight i minus 5 straight j plus 2 straight k

Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah

open vertical bar x with rightwards arrow on top close vertical bar equals open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar space dan space x with rightwards arrow on top equals fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction times v with rightwards arrow on top

maka, panjang proyeksi p terhadap vektor q adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar q with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 1 squared plus left parenthesis negative 5 right parenthesis squared plus 2 to the power of 2 end exponent end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 2 minus 20 plus 6 over denominator square root of 1 plus 25 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 12 over denominator square root of 30 end fraction close vertical bar end cell row blank equals cell 2 over 5 square root of 30 end cell end table

dan proyeksi vektor p terhadap vektor q adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar q with rightwards arrow on top close vertical bar squared end fraction times q with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 1 squared plus left parenthesis negative 5 right parenthesis squared plus 2 squared end root squared end fraction times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses end cell row blank equals cell fraction numerator 2 minus 20 plus 6 over denominator 30 end fraction times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses end cell row blank equals cell fraction numerator negative 12 over denominator 30 end fraction times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 over 5 end cell row 2 row cell negative 4 over 5 end cell end table close parentheses end cell end table

Oleh karena itu, panjang proyeksi dan vektor preyeksi p terhadap q berturut-turut adalah 2 over 5 square root of 30 space dan space minus 2 over 5 straight i plus 2 straight j minus 4 over 5 straight k.

0

Roboguru

Diketahui  dan . Tentukan panjang proyeksi dan vektor proyeksi b terhadap vektor a.

Pembahasan Soal:

Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah

open vertical bar x with rightwards arrow on top close vertical bar equals open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar space dan space x with rightwards arrow on top equals fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction times v with rightwards arrow on top

maka, panjang proyeksi b terhadap vektor a adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell open vertical bar 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 end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses over denominator square root of 2 squared plus 2 squared plus left parenthesis negative 1 right parenthesis squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 12 minus 6 minus 2 over denominator square root of 4 plus 4 plus 1 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 4 over denominator square root of 9 end fraction close vertical bar end cell row blank equals cell open vertical bar 4 over 3 close vertical bar end cell row blank equals cell 4 over 3 end cell end table

dan proyeksi vektor b terhadap vektor a adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell x with rightwards arrow on top 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 squared end fraction times a with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses over denominator open square brackets square root of 2 squared plus 2 squared plus left parenthesis negative 1 right parenthesis squared end root close square brackets squared end fraction times open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses end cell row blank equals cell 4 over 9 open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 8 over 9 end cell row cell 8 over 9 end cell row cell negative 4 over 9 end cell end table close parentheses end cell end table

Oleh karena itu, panjang proyeksi dan vektor preyeksi b terhadap a berturut-turut adalah 4 over 3 space dan space 8 over 9 straight i plus 8 over 9 straight j minus 4 over 9 straight k.

0

Roboguru

Diketahui  dan . O adalah titik pangkal. Tentukan panjang proyeksi dan vektor proyeksi q ke p.

Pembahasan Soal:

Diketahui P left parenthesis 2 comma space 4 comma space 3 right parenthesis dan Q left parenthesis 1 comma space minus 5 comma space 2 right parenthesis, maka

straight p equals 2 straight i plus 4 straight j plus 3 straight k space dan space straight q equals straight i minus 5 straight j plus 2 straight k

Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah

open vertical bar x with rightwards arrow on top close vertical bar equals open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar space dan space x with rightwards arrow on top equals fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction times v with rightwards arrow on top

maka, panjang proyeksi q terhadap vektor p adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar p with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 2 squared plus 4 squared plus 3 squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 2 minus 20 plus 6 over denominator square root of 4 plus 16 plus 9 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 12 over denominator square root of 29 end fraction close vertical bar end cell row blank equals cell 12 over 29 square root of 29 end cell end table

dan proyeksi vektor q terhadap vektor p adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar p with rightwards arrow on top close vertical bar squared end fraction times p with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 2 squared plus 4 squared plus 3 squared end root squared end fraction times open parentheses table row 2 row 4 row 3 end table close parentheses end cell row blank equals cell fraction numerator 2 minus 20 plus 6 over denominator 29 end fraction times open parentheses table row 2 row 4 row 3 end table close parentheses end cell row blank equals cell fraction numerator negative 12 over denominator 29 end fraction times open parentheses table row 2 row 4 row 3 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 14 over 29 end cell row cell negative 48 over 29 end cell row cell negative 36 over 29 end cell end table close parentheses end cell end table

Oleh karena itu, panjang proyeksi dan vektor preyeksi q terhadap p berturut-turut adalah 12 over 29 square root of 29 space dan space minus 14 over 29 straight i minus 48 over 29 straight j minus 36 over 29 straight k.

0

Roboguru

Jika  merupakan vektor proyeksi  pada , maka panjang vektor  adalah ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight c with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator straight a with rightwards arrow on top. straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses stack 2 straight i with rightwards arrow on top minus 6 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top close parentheses space. space open parentheses 2 straight i with rightwards arrow on top minus straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses over denominator square root of 2 cubed plus open parentheses negative 1 close parentheses squared plus open parentheses 2 close parentheses squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 4 plus 6 plus 8 over denominator square root of 4 plus 1 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 18 over denominator square root of 9 end fraction close vertical bar end cell row blank equals cell open vertical bar 18 over 3 close vertical bar end cell row blank equals 6 end table end style 

1

Roboguru

Pembahasan Soal:

M i s a l k a n space p r o y e k s i space v e k t o r space a with rightwards arrow on top space k e space v e k t o r space b with rightwards arrow on top space a d a l a h space v e k t o r space c with rightwards arrow on top space d a n space p a n j a n g n y a space s a m a space d e n g a n space 1 third p a n j a n g space v e k t o r space space b with rightwards arrow on top comma space m a k a  c with rightwards arrow on top equals 1 third space b with rightwards arrow on top  K a r e n a space  c with rightwards arrow on top equals open parentheses fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator vertical line b with rightwards arrow on top vertical line squared end fraction close parentheses b with rightwards arrow on top  M a k a  fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator vertical line b with rightwards arrow on top vertical line squared end fraction space equals space 1 third  fraction numerator left parenthesis i plus 6 j minus k right parenthesis times left parenthesis 7 i plus 4 j plus m k right parenthesis over denominator vertical line 7 i plus 4 j plus m k vertical line squared end fraction equals 1 third  fraction numerator 1 times 7 plus 6 times 4 plus left parenthesis negative 1 right parenthesis times m over denominator 7 squared plus 4 squared plus m squared end fraction equals 1 third  fraction numerator 7 plus 24 minus m over denominator 49 plus 16 plus m squared end fraction space equals 1 third  fraction numerator 31 minus m over denominator 65 plus m squared end fraction equals 1 third  3 left parenthesis 31 minus m right parenthesis equals 1 left parenthesis 65 plus m squared right parenthesis  93 minus 3 m equals 65 plus m squared m squared plus 3 m plus 65 minus 93 equals 0 m squared plus 3 m minus 28 equals 0 left parenthesis m plus 7 right parenthesis left parenthesis m minus 4 right parenthesis equals 0 S e h i n g g a m equals negative 7 a t a u m equals 4.

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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