Roboguru

Diketahuivektoru=⎝⎛​022​⎠⎞​danv=⎝⎛​−202​⎠⎞​.Proyeksivektororthgonaludanvadalah...

Pertanyaan

D i k e t a h u i space v e k t o r space u with rightwards arrow on top equals open parentheses table row 0 row 2 row 2 end table close parentheses space d a n space v with rightwards arrow on top equals open parentheses table row cell negative 2 end cell row 0 row 2 end table close parentheses. space P r o y e k s i space v e k t o r space o r t h g o n a l stack space u with rightwards arrow on top space d a n space stack v space with rightwards arrow on top a d a l a h...

  1. -i + k

  2. negative i plus 1 half k

  3. -i - k

  4. -2i + k

  5. 2i - k

Pembahasan Soal:

P r o y e k s i space v e k t o r space o r t h o g o n a l space u space ⃗ space p a d a space v space ⃑ comma space m i s a l k a n space c space ⃗  c ⃗ equals fraction numerator u ⃑. v ⃑ over denominator vertical line v ⃑ vertical line squared end fraction. v ⃑ equals fraction numerator open parentheses table row 0 row 2 row 2 end table close parentheses. open parentheses table row cell negative 2 end cell row 0 row 2 end table close parentheses over denominator left parenthesis negative 2 right parenthesis squared plus left parenthesis 0 right parenthesis squared plus left parenthesis 2 right parenthesis squared end fraction open parentheses table row cell negative 2 end cell row 0 row 2 end table close parentheses  equals fraction numerator left parenthesis 0 right parenthesis. left parenthesis negative 2 right parenthesis plus left parenthesis 2 right parenthesis. left parenthesis 0 right parenthesis plus 2 left parenthesis 2 right parenthesis over denominator 8 end fraction open parentheses table row cell negative 2 end cell row 0 row 2 end table close parentheses  equals 4 over 8 open parentheses table row cell negative 2 end cell row 0 row 2 end table close parentheses  c space ⃗ equals 1 half open parentheses table row cell negative 2 end cell row 0 row 2 end table close parentheses equals open parentheses table row cell negative 1 end cell row 0 row 1 end table close parentheses space a t a u space c space ⃗ equals negative i plus k

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Kurnia

Mahasiswa/Alumni Universitas Jember

Terakhir diupdate 04 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahuivektoru=⎝⎛​7−41​⎠⎞​danv=⎝⎛​−2−10​⎠⎞​.Proyeksivektororthogonaludanvadalah...

Pembahasan Soal:

P r o y e k s i space v e k t o r space u space ⃗ space p a d a space v space ⃗ space a d a l a h semicolon  fraction numerator u ⃗. v ⃗ over denominator vertical line v ⃗ vertical line squared end fraction. v ⃗  M a k a semicolon  fraction numerator u ⃗. v ⃗ over denominator vertical line v ⃗ vertical line squared end fraction. v ⃗ equals fraction numerator open parentheses table row 7 row cell negative 4 end cell row 1 end table close parentheses open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 0 end table close parentheses over denominator open parentheses square root of open parentheses negative 2 close parentheses squared plus open parentheses negative 1 close parentheses squared plus 0 squared end root close parentheses end fraction. open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 0 end table close parentheses  equals open parentheses fraction numerator negative 14 plus 4 plus 0 over denominator open parentheses square root of 5 close parentheses squared end fraction close parentheses. open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 0 end table close parentheses  equals open parentheses table row 4 row 2 row 0 end table close parentheses

0

Roboguru

Diketahui proyeksi skalar u→ pada v adalah −3. Jika  dan , maka vektor proyeksi u pada w adalah ....

Pembahasan Soal:

Perhatikan bahwa proyeksi skalar undefined pada undefined adalah begin mathsize 14px style negative sign 3 end style. Oleh karena itu, didapat

begin mathsize 14px style fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction equals minus sign 3 end style 

Perhatikan bahwa begin mathsize 14px style italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top end style pasti bernilai negatif karena begin mathsize 14px style italic v with rightwards arrow on top end style tidak mungkin bernilai negatif.

Jadi, sudut antara undefined dan undefined pasti membentuk sudut tumpul seperti digambarkan sebagai berikut.

Misalkan vektor proyeksi undefined pada undefined adalah begin mathsize 14px style italic p with rightwards arrow on top end style maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell italic p with rightwards arrow on top end cell equals cell open parentheses fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar squared end fraction close parentheses italic v with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar ⋅ open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses italic v with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses open parentheses fraction numerator italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses end cell row blank equals cell negative sign 3 open parentheses fraction numerator italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close parentheses italic v with rightwards arrow on top end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator square root of open parentheses negative sign 7 close parentheses squared plus open parentheses negative sign 4 close parentheses squared plus 4 squared end root end fraction close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator square root of 49 plus 16 plus 16 end root end fraction close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator square root of 81 end fraction close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign 3 over 9 close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign 1 third close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses table row cell 7 over 3 end cell row cell 4 over 3 end cell row cell negative sign 4 over 3 end cell end table close parentheses end cell row blank equals cell 7 over 3 italic i with rightwards arrow on top plus 4 over 3 italic j with rightwards arrow on top minus sign 4 over 3 italic k with rightwards arrow on top end cell end table end style 

Selanjutnya, dapat diperhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell italic w with rightwards arrow on top end cell equals cell open parentheses table row 14 row 8 row cell negative sign 8 end cell end table close parentheses end cell row blank equals cell negative sign 2 open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell negative sign 2 italic v with rightwards arrow on top end cell end table end style 

Jadi, vektor undefined dan undefined saling sejajar dengan arah yang berlawanan.

Perhatikan gambar berikut!

Dapat diperhatikan bahwa proyeksi vektor undefined pada undefined memiliki landasan yang sama dengan proyeksi vektor undefined pada undefined.

Jadi, proyeksi vektor undefined pada undefined sama saja dengan proyeksi vektor begin mathsize 14px style u with rightwards arrow on top end style pada undefined, yaitu vektor undefined.

Oleh karena itu, proyeksi vektor undefined pada undefined adalah

begin mathsize 14px style italic p with rightwards arrow on top equals 7 over 3 italic i with rightwards arrow on top plus 4 over 3 italic j with rightwards arrow on top minus sign 4 over 3 italic k with rightwards arrow on top end style 

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Diketahui A(4, 0, 0), B(0, -4, 0), dan C(0, 0, 4). Panjang vektor proyeksi AC ke vektor AB 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 A(3, 0, 0), B(0, - 3, 0), dan C(0, 0, 4). Panjang vektor proyeksi AC ke vektor AB 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 stylebegin mathsize 12px style open parentheses a subscript 1 comma space a subscript 2 comma space a subscript 3 close parentheses end style dan begin mathsize 12px style Q with rightwards arrow on top end style = begin 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 P with rightwards arrow on top. Q with rightwards arrow on top equals open parentheses a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3 close parentheses  open vertical bar P with rightwards arrow on top close vertical bar equals square root of open parentheses a subscript 1 close parentheses squared plus open parentheses a subscript 2 close parentheses squared plus open parentheses a subscript 3 close parentheses squared end root end style

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

begin mathsize 12px style Z with rightwards arrow on top equals fraction numerator P with rightwards arrow on top. Q with rightwards arrow on top over denominator stack open vertical bar Q close vertical bar with rightwards arrow on top end fraction space space end style

 

Diketahui A(3, 0, 0), B(0, - 3, 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 begin mathsize 12px style stack A B with rightwards arrow on top end style, yaitu:

begin mathsize 12px style stack A C with rightwards arrow on top equals C minus A equals open parentheses table row 0 row 0 row 4 end table close parentheses minus open parentheses table row 3 row 0 row 0 end table close parentheses equals open parentheses table row cell negative 3 end cell row 0 row 4 end table close parentheses  stack A B with rightwards arrow on top equals B minus A equals open parentheses table row 0 row cell negative 3 end cell row 0 end table close parentheses minus open parentheses table row 3 row 0 row 0 end table close parentheses equals open parentheses table row cell negative 3 end cell row cell negative 3 end cell row 0 end table close parentheses end style

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

begin mathsize 12px style Z with rightwards arrow on top equals fraction numerator stack A C with rightwards arrow on top. stack A B with rightwards arrow on top over denominator open vertical bar stack A B with rightwards arrow on top close vertical bar end fraction  Z with rightwards arrow on top equals fraction numerator open parentheses table row cell negative 3 end cell row 0 row 4 end table close parentheses. open parentheses table row cell negative 3 end cell row cell negative 3 end cell row 0 end table close parentheses over denominator square root of open parentheses negative 3 close parentheses squared plus open parentheses negative 3 close parentheses squared plus 0 squared end root end fraction  Z with rightwards arrow on top equals fraction numerator 9 plus 0 plus 0 over denominator square root of 18 end fraction  Z with rightwards arrow on top equals fraction numerator 9 over denominator 3 square root of 2 end fraction  Z with rightwards arrow on top equals fraction numerator 3 over denominator square root of 2 end fraction  Z with rightwards arrow on top equals 3 over 2 square root of 2 end style

Jadi, panjang vektor begin mathsize 12px style stack A C with rightwards arrow on top end style ke begin mathsize 12px style stack A B with rightwards arrow on top end style adalah begin mathsize 12px style 3 over 2 square root of 2 end style

0

Roboguru

Diketahui vektor a=2i−2pj​+4k dan b=i−3j​+4k. Jika panjang proyeksi vektor a pada b adalah 26​6​, nilai p = ....

Pembahasan Soal:

begin mathsize 14px style open vertical bar straight c with rightwards arrow on top close vertical bar equals fraction numerator straight a with rightwards arrow on top straight space. straight space straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction  fraction numerator 6 over denominator square root of 26 end fraction equals fraction numerator open parentheses table row 2 row cell negative 2 straight p end cell row 4 end table close parentheses. open parentheses table row 1 row cell negative 3 end cell row 4 end table close parentheses over denominator square root of 1 squared plus open parentheses negative 3 close parentheses squared plus 4 squared end root end fraction  fraction numerator 6 over denominator square root of 26 end fraction equals fraction numerator 2 plus 6 straight p plus 16 over denominator square root of 26 end fraction    Maka  6 equals 2 plus 6 straight p plus 16  6 straight p equals 6 minus 2 minus 16  6 straight p equals negative 12  straight p equals negative 2 end style

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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