Roboguru

Diketahui: ,  Tentukan: a. 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:

a. Panjang proyeksi vektor dari a ke b

Pembahasan Soal:

Diperoleh hasil panjang ooyeksi yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a subscript b close vertical bar end cell equals cell fraction numerator open vertical bar a times b close vertical bar over denominator open vertical bar b 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 6 squared plus open parentheses negative 3 close parentheses squared plus 2 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 36 plus 9 plus 4 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 49 end fraction end cell row blank equals cell 4 over 7 end cell end table

Dengan demikian, panjang proyeksi vektor dari a ke b adalah 4 over 7.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Hermawan

Mahasiswa/Alumni Universitas Lampung

Terakhir diupdate 24 September 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

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

Diketahui  maka panjang proyeksi  pada  adalah ...

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator a times b over denominator open vertical bar b close vertical bar end fraction end cell equals cell fraction numerator 2 times 1 plus 2 times left parenthesis negative 2 right parenthesis plus left parenthesis negative 4 right parenthesis times 2 over denominator square root of 1 squared plus left parenthesis negative 2 right parenthesis squared plus 2 to the power of 2 end exponent end root end fraction end cell row blank equals cell fraction numerator 2 plus open parentheses negative 4 close parentheses plus open parentheses negative 8 close parentheses over denominator square root of 9 end fraction end cell row blank equals cell negative 10 over 3 end cell end table end style
 

Panjang proyeksi begin mathsize 14px style a end style pada begin mathsize 14px style b end style adalah begin mathsize 14px style negative 10 over 3 end style 

1

Roboguru

Jika vektor  dan  panjang vektor  adalah ...

Pembahasan Soal:

Tentukan terlebih dahulu hasil operasi hitung vektornya, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 a with rightwards harpoon with barb upwards on top plus 3 b with rightwards harpoon with barb upwards on top end cell equals cell 2 open parentheses table row cell negative 3 end cell row 0 row 5 end table close parentheses plus 3 open parentheses table row 4 row 1 row cell negative 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 6 end cell row 0 row 10 end table close parentheses plus open parentheses table row 12 row 3 row cell negative 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 6 plus 12 end cell row cell 0 plus 3 end cell row cell 10 minus 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row 6 row 3 row 4 end table close parentheses end cell end table

Kemudian diperoleh panjang vektor yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar 2 a with rightwards harpoon with barb upwards on top plus 3 b with rightwards harpoon with barb upwards on top close vertical bar end cell equals cell square root of 6 squared plus 3 squared plus 4 squared end root end cell row blank equals cell square root of 36 plus 9 plus 16 end root end cell row blank equals cell square root of 61 end cell end table

Maka, panjang vektor 2 a with rightwards harpoon with barb upwards on top plus 3 b with rightwards harpoon with barb upwards on top adalah square root of 61.

0

Roboguru

Diketahui vector  dan vector . Jika panjang proyeksi  pada  adalah , maka nilai dari  yang memenuhi adalah ....

Pembahasan Soal:

Panjang proyeksi vektor b with rightwards harpoon with barb upwards on top pada a with rightwards harpoon with barb upwards on top, yaitu open vertical bar p close vertical bar dapat kita rumuskan:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open vertical bar p close vertical bar end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator open vertical bar a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top close vertical bar over denominator open vertical bar a with rightwards harpoon with barb upwards on top close vertical bar end fraction end cell end table 

dengan open vertical bar a with rightwards harpoon with barb upwards on top close vertical bar equals panjan g space a with rightwards harpoon with barb upwards on top dan open vertical bar a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top close vertical bar equals nilai space mutlak space dari space a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top 

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar p close vertical bar end cell equals 13 row cell fraction numerator open vertical bar a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top close vertical bar over denominator open vertical bar a with rightwards harpoon with barb upwards on top close vertical bar end fraction end cell equals 13 row cell fraction numerator open vertical bar 3 times x plus 4 times 1 plus 12 times 3 close vertical bar over denominator square root of 3 squared plus 4 squared plus 12 squared end root end fraction end cell equals 13 row cell fraction numerator open vertical bar 3 x plus 4 plus 36 close vertical bar over denominator square root of 9 plus 16 plus 144 end root end fraction end cell equals 13 row cell fraction numerator open vertical bar 3 x plus 40 close vertical bar over denominator square root of 169 end fraction end cell equals 13 row cell fraction numerator open vertical bar 3 x plus 40 close vertical bar over denominator 13 end fraction end cell equals 13 row cell open vertical bar 3 x plus 40 close vertical bar end cell equals cell 13 times 13 end cell row cell 3 x plus 40 end cell equals 169 row cell 3 x end cell equals cell 169 minus 40 end cell row cell 3 x end cell equals 129 row x equals cell 129 over 3 end cell row x equals 43 end table     

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Diketahui titik . Panjang proyeksi vektor  pada  sama dengan...

Pembahasan Soal:

Ingat kembali panjang proyeksi berikut.

  • Jika vektor top enclose a diproyeksikan secara orthogonal pada top enclose b, maka panjang proyeksi vektornya adalah 

open vertical bar top enclose c close vertical bar equals open vertical bar fraction numerator top enclose a bullet top enclose b over denominator open vertical bar top enclose b close vertical bar end fraction close vertical bar 

             dengan

top enclose a bullet top enclose b equals a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3 space dan space open vertical bar top enclose b close vertical bar equals square root of b subscript 1 squared plus b subscript 2 squared plus b subscript 3 squared end root   

Dari rumus di atas, diperoleh perhitungan sebagai berikut.

Menentukan top enclose AB  

table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose AB end cell equals cell straight B minus straight A end cell row blank equals cell left parenthesis 5 comma space 2 comma space 7 right parenthesis minus left parenthesis 3 comma space 2 comma space 4 right parenthesis end cell row blank equals cell left parenthesis 5 minus 3 comma space 2 minus 2 comma space 7 minus 4 right parenthesis end cell row blank equals cell left parenthesis 2 comma space 0 comma space 3 right parenthesis end cell end table   

Menentukan top enclose AC   

table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose A C end enclose end cell equals cell C minus straight A end cell row blank equals cell left parenthesis 2 comma space 0 comma space 6 right parenthesis minus left parenthesis 3 comma space 2 comma space 4 right parenthesis end cell row blank equals cell left parenthesis 2 minus 3 comma space 0 minus 2 comma space 6 minus 4 right parenthesis end cell row blank equals cell left parenthesis negative 1 comma space minus 2 comma space 2 right parenthesis end cell end table    

Panjang proyeksi  top enclose AB pada top enclose AC adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose d end cell equals cell open vertical bar fraction numerator top enclose AB bullet top enclose AC over denominator open vertical bar top enclose AC close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator left parenthesis 2 comma space 0 comma space 3 right parenthesis bullet left parenthesis negative 1 comma space minus 2 comma space 2 right parenthesis over denominator square root of left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared plus 2 squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 2 left parenthesis negative 1 right parenthesis plus 0 left parenthesis negative 2 right parenthesis plus 3 left parenthesis 2 right parenthesis over denominator square root of 1 plus 4 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 2 plus 0 plus 6 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 row blank equals cell 1 1 third end cell end table  

Jadi, panjang proyeksi  top enclose AB pada top enclose AC adalah 1 1 third.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

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