Dua buah vektor a dan b (a < b) resultannya adalah R. Bila R = 3a dan sudut antara R dan a adalah 30o, hitung besar sudut apit antara a dan b!

Pertanyaan

Dua buah vektor a dan b (a < b) resultannya adalah R. Bila R = 3a dan sudut antara R dan a adalah 30o, hitung besar sudut apit antara a dan b!

F. Arifin

Master Teacher

Mahasiswa/Alumni Universitas Trisakti

Jawaban terverifikasi

Jawaban

besar sudut apit antara a dan b adalah 43,2o.

Pembahasan

Diketahui :

a with rightwards harpoon with barb upwards on top plus b with rightwards harpoon with barb upwards on top equals R with rightwards harpoon with barb upwards on top R with rightwards arrow on top equals 3 a with rightwards arrow on top angle R with rightwards harpoon with barb upwards on top comma a with rightwards harpoon with barb upwards on top equals 30 to the power of omicron

Ditanya :

angle a with rightwards arrow on top comma b with rightwards arrow on top equals ?

Jawab :

Hubungan a with rightwards arrow on top dan space b with rightwards arrow on top :

b squared equals R squared plus a squared minus 2 R a cos 30 b squared equals open parentheses 3 a close parentheses squared plus a squared minus 2 times 3 a times a times 1 half square root of 3 b squared equals 9 a squared plus a squared minus 3 square root of 3 a squared b squared equals 10 a squared minus 3 square root of 3 a squared b squared equals a squared left parenthesis 10 minus 3 square root of 3 right parenthesis b equals a square root of 10 minus 3 square root of 3 end root

Sudut antara a with rightwards arrow on top space dan space b with rightwards arrow on top :

R squared equals a squared plus b squared plus 2 a b cos theta open parentheses 3 a close parentheses squared equals a squared plus a squared left parenthesis 10 minus 3 square root of 3 right parenthesis plus 2 a times a square root of 10 minus 3 square root of 3 end root cos theta 9 a squared equals a squared plus a squared left parenthesis 10 minus 3 square root of 3 right parenthesis plus 2 a squared square root of 10 minus 3 square root of 3 end root cos theta 9 equals 1 plus 10 minus 3 square root of 3 plus 2 square root of 10 minus 3 square root of 3 end root cos theta 9 minus 1 minus 10 plus 3 square root of 3 equals 2 square root of 10 minus 3 square root of 3 end root cos theta cos theta equals fraction numerator negative 2 plus 3 square root of 3 over denominator 2 square root of 10 minus 3 square root of 3 end root end fraction cos theta equals 0 comma 73 theta equals cos to the power of negative 1 end exponent left parenthesis 0 comma 73 right parenthesis theta equals 43 comma 2 to the power of omicron

Jadi besar sudut apit antara a dan b adalah 43,2o.

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