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Pertanyaan

Vektor-vektor posisi titik A dan B masing-masing relatif terhadap titik asal O adalah bold italic a dan bold italic b. Titik B sedemikian sehingga OP with rightwards harpoon with barb upwards on top equals 4 OB with rightwards harpoon with barb upwards on top. Titik tengah AB with rightwards harpoon with barb upwards on top adalah Q. Titik R sedemikian sehingga OR with rightwards harpoon with barb upwards on top equals 8 over 5 OQ with rightwards harpoon with barb upwards on top. Tentukan dalam bold italic a dan bold italic b, untuk vektor-vektor berikut.

a. OQ with rightwards harpoon with barb upwards on top 

b. OR with rightwards harpoon with barb upwards on top 

c. AR with rightwards harpoon with barb upwards on top 

d. RP with rightwards harpoon with barb upwards on top

e. Tunjukkan bahwa R terletak pada AP with rightwards harpoon with barb upwards on top dan nyatakan perbandingan AR with rightwards harpoon with barb upwards on top divided by RP with rightwards harpoon with barb upwards on top.space 

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

a. Diketahui vektor-vektor posisi titik A dan B masing-masing relatif terhadap titik asal O adalah bold italic a dan bold italic b. Berarti

table attributes columnalign right center left columnspacing 0px end attributes row cell stack O A with rightwards harpoon with barb upwards on top end cell equals bold italic a row cell stack O B with rightwards harpoon with barb upwards on top end cell equals bold italic b end table 

 Karena titik Q adalah titik tengah stack A B with rightwards harpoon with barb upwards on top, maka 

table attributes columnalign right center left columnspacing 0px end attributes row cell stack A Q with rightwards harpoon with barb upwards on top end cell equals cell 1 half stack A B with rightwards harpoon with barb upwards on top end cell row cell stack O Q with rightwards harpoon with barb upwards on top minus stack O A with rightwards harpoon with barb upwards on top end cell equals cell 1 half open parentheses stack O B with rightwards harpoon with barb upwards on top minus stack O A with rightwards harpoon with barb upwards on top close parentheses end cell row cell stack O Q with rightwards harpoon with barb upwards on top minus bold italic a end cell equals cell 1 half open parentheses bold italic b minus bold italic a close parentheses end cell row cell stack O Q with rightwards harpoon with barb upwards on top minus bold italic a end cell equals cell 1 half bold italic b minus 1 half bold italic a end cell row cell stack O Q with rightwards harpoon with barb upwards on top end cell equals cell 1 half bold italic b minus 1 half bold italic a plus bold italic a end cell row cell stack O Q with rightwards harpoon with barb upwards on top end cell equals cell 1 half bold italic b plus 1 half bold italic a end cell end table   

b. Diketahui stack O R with rightwards harpoon with barb upwards on top equals 8 over 5 stack O Q with rightwards harpoon with barb upwards on top, maka diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell stack O R with rightwards harpoon with barb upwards on top end cell equals cell 8 over 5 stack O Q with rightwards harpoon with barb upwards on top end cell row blank equals cell 8 over 5 open parentheses 1 half bold italic b plus 1 half bold italic a close parentheses end cell row blank equals cell 4 over 5 bold italic b plus 4 over 5 bold italic a end cell end table 

c. Diketahui stack O A with rightwards harpoon with barb upwards on top equals bold italic a dan dari poin c didapat stack O R with rightwards harpoon with barb upwards on top equals 4 over 5 bold italic b plus 4 over 5 bold italic a. Maka diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell stack A R with rightwards harpoon with barb upwards on top end cell equals cell stack O R with rightwards harpoon with barb upwards on top minus stack O A with rightwards harpoon with barb upwards on top end cell row blank equals cell open parentheses 4 over 5 bold italic b plus 4 over 5 bold italic a close parentheses minus bold italic a end cell row blank equals cell 4 over 5 bold italic b plus 4 over 5 bold italic a minus bold italic a end cell row blank equals cell 4 over 5 bold italic b plus 4 over 5 bold italic a minus 5 over 5 bold italic a end cell row blank equals cell 4 over 5 bold italic b minus 1 fifth bold italic a end cell end table  

d. Diketahui stack O P with rightwards harpoon with barb upwards on top equals 4 stack O B with rightwards harpoon with barb upwards on top, berarti

table attributes columnalign right center left columnspacing 0px end attributes row cell stack O P with rightwards harpoon with barb upwards on top end cell equals cell 4 stack O B with rightwards harpoon with barb upwards on top end cell row blank equals cell 4 bold italic b end cell end table

Dari poin b diperoleh stack O R with rightwards harpoon with barb upwards on top equals 4 over 5 bold italic b plus 4 over 5 bold italic a, maka diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell stack R P with rightwards harpoon with barb upwards on top end cell equals cell stack O P with rightwards harpoon with barb upwards on top minus stack O R with rightwards harpoon with barb upwards on top end cell row blank equals cell 4 bold italic b minus open parentheses 4 over 5 bold italic a plus 4 over 5 bold italic b close parentheses end cell row blank equals cell 4 bold italic b minus 4 over 5 bold italic a minus 4 over 5 bold italic b end cell row blank equals cell negative 4 over 5 bold italic a plus 4 bold italic b minus 4 over 5 bold italic b end cell row blank equals cell negative bold 4 over bold 5 bold italic a plus 20 over 5 bold italic b minus 4 over 5 bold italic b end cell row blank equals cell negative bold 4 over bold 5 bold italic a plus 16 over 5 bold italic b end cell row blank equals cell 16 over 5 bold italic b minus 4 over 5 bold italic a end cell end table 

e. Akan ditunjukkan bahwa R terletak pada stack A P with rightwards harpoon with barb upwards on top. Apabila R terletak pada stack A P with rightwards harpoon with barb upwards on top, maka harus memenuhi stack R P with rightwards harpoon with barb upwards on top equals k stack A R with rightwards harpoon with barb upwards on top dengan k suatu bilangan positif. Perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell stack R P with rightwards harpoon with barb upwards on top end cell equals cell 16 over 5 bold italic b minus 4 over 5 bold italic a end cell row blank equals cell fraction numerator 4 cross times 4 over denominator 5 end fraction bold italic b minus fraction numerator 4 cross times 1 over denominator 5 end fraction bold italic a end cell row blank equals cell 4 open parentheses 4 over 5 bold italic b minus 1 fifth bold italic a close parentheses end cell row blank equals cell 4 stack A R with rightwards harpoon with barb upwards on top end cell end table 

Ternyata diperoleh bahwa stack R P with rightwards harpoon with barb upwards on top equals k stack A R with rightwards harpoon with barb upwards on top dengan k equals 4. Artinya, terbukti bahwa R terletak pada stack A P with rightwards harpoon with barb upwards on top. Selanjutnya perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell stack R P with rightwards harpoon with barb upwards on top end cell equals cell 4 stack A R with rightwards harpoon with barb upwards on top end cell row cell 1 fourth end cell equals cell fraction numerator stack A R with rightwards harpoon with barb upwards on top over denominator stack R P with rightwards harpoon with barb upwards on top end fraction end cell end table

Sehingga didapat perbandingan stack A R with rightwards harpoon with barb upwards on top divided by stack R P with rightwards harpoon with barb upwards on top adalah 1 divided by 4 

Dengan demikian, diperoleh bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell stack O Q with rightwards harpoon with barb upwards on top end cell equals cell 1 half bold italic b plus 1 half bold italic a end cell row cell stack O R with rightwards harpoon with barb upwards on top end cell equals cell 4 over 5 bold italic b plus 4 over 5 bold italic a end cell row cell stack A R with rightwards harpoon with barb upwards on top end cell equals cell 4 over 5 bold italic b minus 1 fifth bold italic a end cell row cell stack R P with rightwards harpoon with barb upwards on top end cell equals cell 16 over 5 bold italic b minus 4 over 5 bold italic a end cell row cell straight R space terletak space pada space stack A P with rightwards harpoon with barb upwards on top space dengan space stack A R with rightwards harpoon with barb upwards on top divided by stack R P with rightwards harpoon with barb upwards on top end cell equals cell 1 divided by 4 end cell end table

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