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Jika a=i+j​+2k dan b=2i−j​+k, buktikan bahwa a+b adalah tegak lurus pada a−b.

Pertanyaan

Jika a equals i with hat on top plus j with hat on top plus 2 k with hat on top space dan space b equals 2 i with hat on top minus j with hat on top plus k with hat on top, buktikan bahwa a plus b adalah tegak lurus pada a minus b.

S. Nur

Master Teacher

Jawaban terverifikasi

Jawaban

terbukti bahwa a with rightwards arrow on top plus b with rightwards arrow on top tegak lurus pada a with rightwards arrow on top minus b with rightwards arrow on top.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah terbukti bahwa bold italic a with bold rightwards arrow on top bold plus bold italic b with bold rightwards arrow on top tegak lurus pada bold italic a with bold rightwards arrow on top bold minus bold italic b with bold rightwards arrow on top.

Ingat dua vektor u with rightwards arrow on top dan v with rightwards arrow on top saling tegak lurus apabila memenuhi u with rightwards arrow on top times v with rightwards arrow on top equals 0. Lalu, jika terdapat vektor u with rightwards arrow on top equals u subscript 1 i with rightwards arrow on top plus u subscript 2 j with rightwards arrow on top plus u subscript 3 k with rightwards arrow on top dan v with rightwards arrow on top equals v subscript 1 i with rightwards arrow on top plus v subscript 2 j with rightwards arrow on top plus v subscript 3 k with rightwards arrow on top, maka diperoleh u with rightwards arrow on top times v with rightwards arrow on top sebagai berikut.

u with rightwards arrow on top times v with rightwards arrow on top equals u subscript 1 v subscript 1 plus u subscript 2 v subscript 2 plus u subscript 3 v subscript 3 

Diketahui vektor a with rightwards arrow on top equals i with rightwards arrow on top plus j with rightwards arrow on top plus 2 k with rightwards arrow on top dan b with rightwards arrow on top equals 2 i with rightwards arrow on top minus j with rightwards arrow on top plus k with rightwards arrow on top, maka diperoleh a with rightwards arrow on top plus b with rightwards arrow on top dan a with rightwards arrow on top minus b with rightwards arrow on top sebagai berikut.

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 i with rightwards arrow on top plus j with rightwards arrow on top plus 2 k with rightwards arrow on top close parentheses plus open parentheses 2 i with rightwards arrow on top minus j with rightwards arrow on top plus k with rightwards arrow on top close parentheses end cell row blank equals cell i with rightwards arrow on top plus 2 i with rightwards arrow on top plus j with rightwards arrow on top minus j with rightwards arrow on top plus 2 k with rightwards arrow on top plus k with rightwards arrow on top end cell row blank equals cell 3 i with rightwards arrow on top plus 3 k with rightwards arrow on top end cell end table
 

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

Selanjutnya, diperoleh open parentheses a with rightwards arrow on top plus b with rightwards arrow on top close parentheses times open parentheses a with rightwards arrow on top minus b with rightwards arrow on top close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses a with rightwards arrow on top plus b with rightwards arrow on top close parentheses times open parentheses a with rightwards arrow on top minus b with rightwards arrow on top close parentheses end cell equals cell 3 times open parentheses negative 1 close parentheses plus 0 times 2 plus 3 times 1 end cell row blank equals cell negative 3 plus 0 plus 3 end cell row blank equals 0 end table

Berdasarkan perhitungan di atas, diperoleh  open parentheses a with rightwards arrow on top plus b with rightwards arrow on top close parentheses times open parentheses a with rightwards arrow on top minus b with rightwards arrow on top close parentheses equals 0, maka vektor a with rightwards arrow on top plus b with rightwards arrow on top tegak lurus dengan vektor a with rightwards arrow on top minus b with rightwards arrow on top.

Dengan demikian, terbukti bahwa a with rightwards arrow on top plus b with rightwards arrow on top tegak lurus pada a with rightwards arrow on top minus b with rightwards arrow on top.

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