Roboguru
SD

Diketahui a=2i−j​+2k, b=2i+5j​+6k, dan c=−4i+2j​+4k. Tentukan proyeksi skalar dan proyeksi vektor ortogonal (a+b) pada arah vektor (b−c).

Pertanyaan

Diketahui a with rightwards arrow on top equals 2 i with rightwards arrow on top minus j with rightwards arrow on top plus 2 k with rightwards arrow on top comma space b with rightwards arrow on top equals 2 i with rightwards arrow on top plus 5 j with rightwards arrow on top plus 6 k with rightwards arrow on top, dan c with rightwards arrow on top equals negative 4 i with rightwards arrow on top plus 2 j with rightwards arrow on top plus 4 k with rightwards arrow on top. Tentukan proyeksi skalar dan proyeksi vektor ortogonal open parentheses a with rightwards arrow on top plus b with rightwards arrow on top close parentheses pada arah vektor open parentheses b with rightwards arrow on top minus c with rightwards arrow on top close parentheses.

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 52 over 7 comma space 312 over 49 i with rightwards arrow on top plus 156 over 49 j with rightwards arrow on top plus 104 over 49 k with rightwards arrow on top end cell end table

Ingat!

kita punya vektor x with rightwards arrow on top equals x subscript 1 i with rightwards arrow on top plus x subscript 2 j with rightwards arrow on top plus x subscript 3 k with rightwards arrow on top dan y with rightwards arrow on top equals y subscript 1 i with rightwards arrow on top plus y subscript 2 j with rightwards arrow on top plus y subscript 3 k with rightwards arrow on top, rumus proyeksi x with rightwards arrow on top pada y with rightwards arrow on top proyeksi skalar adalah:

fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator open vertical bar y with rightwards arrow on top close vertical bar end fraction 

proyeksi vektor adalah

proyeksi space skalar times fraction numerator straight y with rightwards arrow on top over denominator open vertical bar straight y with rightwards arrow on top close vertical bar end fraction

Langkah 1 mencari open parentheses a with rightwards arrow on top plus b with rightwards arrow on top close parentheses dan open parentheses b with rightwards arrow on top minus c with rightwards arrow on top close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top plus b with rightwards arrow on top end cell equals cell open parentheses table row 2 row cell negative 1 end cell row 2 end table close parentheses plus open parentheses table row 2 row 5 row 6 end table close parentheses end cell row blank equals cell open parentheses table row 4 row 4 row 8 end table close parentheses end cell row blank equals cell 4 i with rightwards arrow on top plus 4 j with rightwards arrow on top plus 8 k with rightwards arrow on top end cell row cell b with rightwards arrow on top minus c with rightwards arrow on top end cell equals cell open parentheses table row 2 row 5 row 6 end table close parentheses minus open parentheses table row cell negative 4 end cell row 2 row 4 end table close parentheses end cell row blank equals cell open parentheses table row 6 row 3 row 2 end table close parentheses end cell row blank equals cell 6 i with rightwards arrow on top plus 3 j with rightwards arrow on top plus 2 k with rightwards arrow on top end cell end table 

Langkah 2 mencari proyeksi vektor dan proyeksi skalar

table attributes columnalign right center left columnspacing 0px end attributes row cell proyeksi space skalar end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator open vertical bar y with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator open parentheses table row 4 row 4 row 8 end table close parentheses open parentheses table row 6 row 3 row 2 end table close parentheses over denominator square root of 6 squared plus 3 squared plus 2 squared end root end fraction end cell row blank equals cell fraction numerator 24 plus 12 plus 16 over denominator square root of 49 end fraction end cell row blank equals cell 52 over 7 end cell row cell proyeksi space vektor end cell equals cell proyeksi space skalar times fraction numerator straight y with rightwards arrow on top over denominator open vertical bar straight y with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell 52 over 7 times fraction numerator open parentheses table row 6 row 3 row 2 end table close parentheses over denominator square root of 49 end fraction end cell row blank equals cell 52 over 49 open parentheses table row 6 row 3 row 2 end table close parentheses end cell row blank equals cell 312 over 49 i with rightwards arrow on top plus 156 over 49 j with rightwards arrow on top plus 104 over 49 k with rightwards arrow on top end cell end table   

dengan demikian, proyeksi skalar dan proyeksi ortogonal adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 52 over 7 comma space 312 over 49 i with rightwards arrow on top plus 156 over 49 j with rightwards arrow on top plus 104 over 49 k with rightwards arrow on top end cell end table

43

3.6 (3 rating)

Adella Rheina Sweeta

Jawaban tidak sesuai

Pertanyaan serupa

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

211

5.0

Jawaban terverifikasi

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