Misalkan titik O(0,0,0) bertindak sebagai titik pangkal. Tentukan besar sudut POQ untuk tiap pasang titik P dan titik Q berikut. c. P(1,2,5) dan Q(−2,1,21​30​)

Pertanyaan

Misalkan titik straight O left parenthesis 0 comma 0 comma 0 right parenthesis bertindak sebagai titik pangkal. Tentukan besar sudut POQ untuk tiap pasang titik straight P dan titik straight Q berikut.

c. straight P left parenthesis 1 comma 2 comma 5 right parenthesis dan straight Q left parenthesis negative 2 comma 1 comma 1 half square root of 30 right parenthesis 

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

besar sudut POQ adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 45 degree end cell end table.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold 45 bold degree end cell end table.

Ingat, untuk menentukan sudut antara dua vektor diperlukan beberapa rumus berikut. 

Jika diketahui dua buah titik straight A left parenthesis x subscript 1 comma y subscript 1 comma z subscript 1 right parenthesis dan straight B left parenthesis x subscript 2 comma y subscript 2 comma z subscript 2 right parenthesis. Maka :

  • AB with rightwards arrow on top equals OB with rightwards arrow on top minus OA with rightwards arrow on top equals open parentheses table row cell x subscript 2 minus x subscript 1 end cell row cell y subscript 2 minus y subscript 1 end cell row cell z subscript 2 minus z subscript 1 end cell end table close parentheses
  • open vertical bar AB with rightwards arrow on top close vertical bar equals square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared plus open parentheses z subscript 2 minus z subscript 1 close parentheses squared end root
  • a with rightwards arrow on top times b with rightwards harpoon with barb upwards on top equals open parentheses x subscript 1 times x subscript 2 close parentheses plus open parentheses y subscript 1 times y subscript 2 close parentheses plus open parentheses z subscript 1 times z subscript 2 close parentheses 
  • cos theta equals 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 open vertical bar b with rightwards arrow on top close vertical bar end fraction 

Pada soal ditanyakan besar sudut POQ dengan titik straight P left parenthesis 1 comma 2 comma 5 right parenthesis dan straight Q left parenthesis negative 2 comma 1 comma 1 half square root of 30 right parenthesis. Titik straight O left parenthesis 0 comma 0 comma 0 right parenthesis bertindak sebagai titik pangkal. Besar sudut POQ dapat kita tentukan dengan perhitungan berikut. 

  • Menentukan vektor OP with rightwards arrow on top dan OQ with rightwards arrow on top

OP with rightwards arrow on top equals open parentheses table row cell 1 minus 0 end cell row cell 2 minus 0 end cell row cell 5 minus 0 end cell end table close parentheses equals open parentheses table row 1 row 2 row 5 end table close parentheses 

OQ with rightwards arrow on top equals open parentheses table row cell negative 2 minus 0 end cell row cell 1 minus 0 end cell row cell 1 half square root of 30 minus 0 end cell end table close parentheses equals open parentheses table row cell negative 2 end cell row 1 row cell 1 half square root of 30 end cell end table close parentheses 

  • Menentukan besar sudut POQ

table attributes columnalign right center left columnspacing 0px end attributes row cell cos theta end cell equals cell fraction numerator OP with rightwards arrow on top times OQ with rightwards arrow on top over denominator open vertical bar OP with rightwards arrow on top close vertical bar open vertical bar OQ with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator open parentheses table row 1 row 2 row 5 end table close parentheses open parentheses table row cell negative 2 end cell row 1 row cell begin display style 1 half end style square root of 30 end cell end table close parentheses over denominator open parentheses square root of 1 squared plus 2 squared plus 5 squared end root close parentheses open parentheses square root of left parenthesis negative 2 right parenthesis squared plus 1 squared plus left parenthesis begin display style 1 half end style square root of 30 right parenthesis squared end root close parentheses end fraction end cell row blank equals cell fraction numerator negative 2 plus 2 plus begin display style 5 over 2 end style square root of 30 over denominator open parentheses square root of 1 plus 4 plus 25 end root close parentheses open parentheses square root of 4 plus 1 plus begin display style 30 over 4 end style end root close parentheses end fraction end cell row blank equals cell fraction numerator begin display style 5 over 2 end style square root of 30 over denominator square root of 30 square root of 5 plus begin display style 30 over 4 end style end root end fraction end cell row blank equals cell fraction numerator begin display style 5 over 2 end style square root of 30 over denominator square root of 30 open parentheses 5 plus 30 over 4 close parentheses end root end fraction end cell row blank equals cell fraction numerator begin display style 5 over 2 end style square root of 30 over denominator square root of 30 open parentheses begin display style fraction numerator 20 plus 30 over denominator 4 end fraction end style close parentheses end root end fraction end cell row blank equals cell fraction numerator begin display style 5 over 2 end style square root of 30 over denominator square root of 30 open parentheses begin display style 50 over 4 end style close parentheses end root end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 5 square root of 30 over denominator 2 end fraction end style over denominator begin display style fraction numerator square root of 30 square root of 50 over denominator square root of 4 end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 5 square root of 30 over denominator 2 end fraction end style over denominator begin display style fraction numerator square root of 30 square root of 50 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator 5 square root of 30 over denominator 2 end fraction cross times fraction numerator 2 over denominator square root of 30 square root of 50 end fraction end cell row blank equals cell fraction numerator 5 over denominator square root of 50 end fraction cross times fraction numerator square root of 50 over denominator square root of 50 end fraction end cell row blank equals cell fraction numerator 5 square root of 50 over denominator 50 end fraction end cell row blank equals cell fraction numerator square root of 25 times 2 end root over denominator 10 end fraction end cell row blank equals cell fraction numerator 5 square root of 2 over denominator 10 end fraction end cell row blank equals cell fraction numerator square root of 2 over denominator 2 end fraction end cell row theta equals cell arc space cos space 1 half square root of 2 end cell row theta equals cell 45 degree end cell end table 

Dengan demikian, besar sudut POQ adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 45 degree end cell end table.

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