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Jika koordinat titik ,  dan , nilai  adalah ....

Pertanyaan

Jika koordinat titik begin mathsize 14px style straight A left parenthesis straight m comma space minus 5 comma space 2 straight n right parenthesis end stylebegin mathsize 14px style straight B left parenthesis straight n comma space 2 straight m comma space straight m plus 2 right parenthesis end style dan begin mathsize 14px style stack A B with rightwards harpoon with barb upwards on top equals open parentheses table row cell negative 7 end cell row cell 1 minus 4 n end cell row cell 3 m end cell end table close parentheses end style, nilai begin mathsize 14px style m end style adalah ....

  1. begin mathsize 14px style 4 end style 

  2. 3

  3. 1

  4. size 14px minus size 14px 3 

  5. size 14px minus size 14px 4 

Pembahasan Soal:

Diketahui

Koordinat titik begin mathsize 14px style straight A left parenthesis straight m comma space minus 5 comma space 2 straight n right parenthesis end stylebegin mathsize 14px style straight B left parenthesis straight n comma space 2 straight m comma space straight m plus 2 right parenthesis end style dan begin mathsize 14px style stack A B with rightwards harpoon with barb upwards on top equals open parentheses table row cell negative 7 end cell row cell 1 minus 4 n end cell row cell 3 m end cell end table close parentheses end style,

Ditanya:

Nilai begin mathsize 14px style m end style.

Dijawab.

Pada soal diketahui bahwa koordinat titik begin mathsize 14px style straight A left parenthesis straight m comma space minus 5 comma space 2 straight n right parenthesis end stylebegin mathsize 14px style straight B left parenthesis straight n comma space 2 straight m comma space straight m plus 2 right parenthesis end style maka kita dapat menentukan vektor kolom begin mathsize 14px style stack A B with rightwards harpoon with barb upwards on top end style yaitu: 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell AB with rightwards harpoon with barb upwards on top end cell equals cell open parentheses table row cell straight b subscript 1 minus straight a subscript 1 end cell row cell straight b subscript 2 minus straight a subscript 2 end cell row cell straight b subscript 3 minus straight a subscript 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell straight n minus straight m end cell row cell 2 straight m minus left parenthesis negative 5 right parenthesis end cell row cell straight m plus 2 minus 2 straight n end cell end table close parentheses end cell row blank equals cell open parentheses table row cell straight n minus straight m end cell row cell 2 straight m plus 5 end cell row cell straight m minus 2 straight n plus 2 end cell end table close parentheses end cell end table end style 

Dengan menggunakan kesamaan Vektor. maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell negative 7 end cell row cell 1 minus 4 straight n end cell row cell 3 straight m end cell end table close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell straight n minus straight m end cell row cell 2 straight m plus 5 end cell row cell straight m minus 2 straight n plus 2 end cell end table close parentheses end cell end table end style 

begin mathsize 14px style straight i right parenthesis space straight n minus straight m equals negative 7 ii right parenthesis space 2 straight m plus 5 equals 1 minus 4 straight n iii right parenthesis space straight m minus 2 straight n plus 2 equals 3 straight m end style  

Dari persamaan i dan ii, kita eliminasi n, maka diperoleh penyelesaian sebagai berikut:

begin mathsize 14px style table row cell straight n minus straight m equals negative 7 rightwards arrow end cell cell 4 straight n minus 4 straight m equals negative 28 end cell row cell 2 straight m plus 5 equals 1 minus 4 straight n rightwards arrow end cell cell 4 straight n plus 2 straight m equals negative 4 minus end cell row blank cell negative 6 straight m equals negative 24 end cell row blank cell straight m equals 4 end cell end table end style 

Jadi, Jika Koordinat titik begin mathsize 14px style straight A left parenthesis straight m comma space minus 5 comma space 2 straight n right parenthesis end stylebegin mathsize 14px style straight B left parenthesis straight n comma space 2 straight m comma space straight m plus 2 right parenthesis end style dan begin mathsize 14px style stack A B with rightwards harpoon with barb upwards on top equals open parentheses table row cell negative 7 end cell row cell 1 minus 4 n end cell row cell 3 m end cell end table close parentheses end style maka nilai begin mathsize 14px style m equals 4 end style.

Oleh karena itu, Jawaban yang benar adalah A

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. RGFLLIMA

Terakhir diupdate 30 April 2021

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Pertanyaan yang serupa

Tentukan modulus dari vektor-vektor berikut. b.  dengan titik  dan titik B

Pembahasan Soal:

Vektor begin mathsize 14px style stack A B with rightwards arrow on top end style dapat ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack A B with rightwards arrow on top end cell equals cell B minus A end cell row blank equals cell open parentheses table row 2 row 1 row cell negative 4 end cell end table close parentheses minus open parentheses table row cell negative 2 end cell row 3 row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row 4 row cell negative 2 end cell row cell negative 3 end cell end table close parentheses end cell end table end style

Modulus vektor tersebut dapat ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack A B with rightwards arrow on top close vertical bar end cell equals cell square root of x squared plus y squared plus z squared end root end cell row blank equals cell square root of 4 squared plus open parentheses negative 2 close parentheses squared plus open parentheses negative 3 close parentheses squared end root end cell row blank equals cell square root of 16 plus 4 plus 9 end root end cell row blank equals cell square root of 29 end cell end table end style

Dengan demikian, modulus vektor begin mathsize 14px style stack A B with rightwards arrow on top end style adalah begin mathsize 14px style square root of 29 end style 

0

Roboguru

Tentukan modulus dari vektor-vektor berikut. b.    dengan titik  dan titik .

Pembahasan Soal:

begin mathsize 14px style stack A B with rightwards arrow on top equals b with rightwards arrow on top minus a with rightwards arrow on top end style 

     begin mathsize 14px style equals open parentheses table row 2 row 1 row cell negative 4 end cell end table close parentheses minus open parentheses table row cell negative 2 end cell row 3 row cell negative 1 end cell end table close parentheses equals open parentheses table row 4 row cell negative 2 end cell row cell negative 3 end cell end table close parentheses end style 

Modulus artinya panjang vektor.

begin mathsize 14px style open vertical bar stack A B with rightwards arrow on top close vertical bar equals square root of 4 squared plus open parentheses negative 2 close parentheses squared plus open parentheses negative 3 close parentheses squared space end root equals square root of 29 end style 

Jadi modulus vektor AB adalah 29.

0

Roboguru

Tentukan modulus dari vektor-vektor berikut. b.  dengan titik  dan titik .

Pembahasan Soal:

Modulus vektor adalah besar atau nilai vektor. Jika diketahui titik begin mathsize 14px style straight A open parentheses x subscript 1 comma space y subscript 1 comma space z subscript 1 close parentheses end style dan titik begin mathsize 14px style straight B open parentheses x subscript 2 comma space y subscript 2 comma space z subscript 2 close parentheses end style, maka besar nilai vektor begin mathsize 14px style AB with rightwards arrow on top end style adalah 

begin mathsize 14px style AB with rightwards arrow on top 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 end style .

Akan ditentukan modulus dari begin mathsize 14px style AB with rightwards arrow on top end style dengan titik begin mathsize 14px style straight A open parentheses negative 2 comma space 3 comma space minus 1 close parentheses end style dan titik begin mathsize 14px style straight B open parentheses 2 comma space 1 comma space minus 4 close parentheses end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell AB with rightwards arrow on top end cell equals cell square root of open parentheses 2 minus open parentheses negative 2 close parentheses close parentheses squared plus open parentheses 1 minus 3 close parentheses squared plus open parentheses negative 4 minus open parentheses negative 1 close parentheses close parentheses squared end root end cell row blank equals cell square root of 4 squared plus open parentheses negative 2 close parentheses squared plus open parentheses negative 3 close parentheses squared end root end cell row blank equals cell square root of 16 plus 4 plus 9 end root end cell row cell AB with rightwards arrow on top end cell equals cell square root of 29 end cell end table end style  

Jadi, diperoleh modulus dari vektor tersebut adalah begin mathsize 14px style square root of 29 end style.

0

Roboguru

Diketahui A(4, 0, 0), B(0, -4, 0), dan C(0, 0, 4). Panjang vektor proyeksi  ke adalah ....

Pembahasan Soal:

Ingat!

Jika titik P(a1, a2, a3) dan Q(b1, b2, b3), maka:

begin mathsize 14px style stack P Q with rightwards arrow on top end style = Q - P = begin mathsize 14px style open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell row cell b subscript 3 end cell end table close parentheses minus open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell row cell a subscript 3 end cell end table close parentheses end style

Misal: begin mathsize 14px style P with rightwards arrow on top end style = (a1, a2, a3) dan begin mathsize 14px style Q with rightwards arrow on top end style = (b1, b2, b3), maka:

size 14px P with size 14px rightwards arrow on top size 14px. size 14px Q with size 14px rightwards arrow on top = (a1b1 + a2b2 + a3b3)

begin mathsize 14px style open vertical bar P with rightwards arrow on top close vertical bar space end style= begin mathsize 14px style square root of open parentheses a subscript 1 close parentheses squared plus open parentheses a subscript 2 close parentheses squared plus open parentheses a subscript 3 close parentheses squared end root end style

Misal begin mathsize 14px style Z with rightwards arrow on top end style adalah Proyeksi skalar vektor begin mathsize 14px style P with rightwards arrow on top end style pada vektor begin mathsize 14px style Q with rightwards arrow on top end style, maka:

begin mathsize 14px style Z with rightwards arrow on top end style = begin mathsize 14px style fraction numerator P with rightwards arrow on top. Q with rightwards arrow on top over denominator open vertical bar Q with rightwards arrow on top close vertical bar end fraction end style

Diketahui titik A(4, 0, 0), B(0, –4, 0) dan C(0, 0, 4), maka panjang vektor proyeksi begin mathsize 14px style stack A C with rightwards arrow on top end style ke begin mathsize 14px style stack A B with rightwards arrow on top end style, yaitu:

begin mathsize 14px style stack A C with rightwards arrow on top end style = C - A = begin mathsize 14px style open parentheses table row 0 row 0 row 4 end table close parentheses minus open parentheses table row 4 row 0 row 0 end table close parentheses equals open parentheses table row cell negative 4 end cell row 0 row 4 end table close parentheses end style

begin mathsize 14px style stack A B with rightwards arrow on top end style = B - A = begin mathsize 14px style open parentheses table row 0 row cell negative 4 end cell row 0 end table close parentheses minus open parentheses table row 4 row 0 row 0 end table close parentheses equals open parentheses table row cell negative 4 end cell row cell negative 4 end cell row 0 end table close parentheses end style

Misal begin mathsize 14px style Z with rightwards arrow on top end style adalah proyeksi vektor begin mathsize 14px style stack A C with rightwards arrow on top end style ke begin mathsize 14px style stack A B with rightwards arrow on top end style, maka:

begin mathsize 14px style Z with rightwards arrow on top end style = begin mathsize 14px style fraction numerator stack A C with rightwards arrow on top. stack A B with rightwards arrow on top over denominator open vertical bar stack A B with rightwards arrow on top close vertical bar end fraction equals fraction numerator open parentheses table row cell negative 4 end cell row 0 row 4 end table close parentheses. open parentheses table row cell negative 4 end cell row cell negative 4 end cell row 0 end table close parentheses over denominator square root of open parentheses negative 4 close parentheses squared plus open parentheses negative 4 close parentheses squared plus 0 squared end root end fraction end style

   = begin mathsize 14px style fraction numerator 16 plus 0 plus 0 over denominator square root of 32 end fraction equals fraction numerator 16 over denominator 4 square root of 2 end fraction equals fraction numerator 4 over denominator square root of 2 end fraction equals 2 square root of 2 end style

Jadi, panjang vektor proyeksi begin mathsize 14px style stack A C with rightwards arrow on top end style ke begin mathsize 14px style stack A B with rightwards arrow on top end style adalah begin mathsize 14px style 2 square root of 2 end style

0

Roboguru

Jarak antara titik dan adalah ...

Pembahasan Soal:

Ingat rumus menentukan panjang vektor untuk vektor pada dimensi tiga yaitu a with rightwards arrow on top equals square root of x squared plus y squared plus z squared end root.

Menghitung vektor PQ with rightwards arrow on top

table attributes columnalign right center left columnspacing 0px end attributes row cell PQ with rightwards arrow on top end cell equals cell straight Q minus straight P end cell row blank equals cell open parentheses negative 1 , 2 comma 3 close parentheses minus open parentheses 2 , 8 comma negative 3 close parentheses end cell row blank equals cell open parentheses negative 3 comma negative 6 , 6 close parentheses end cell end table

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell PQ with rightwards arrow on top end cell equals cell square root of x squared plus y squared plus z squared end root end cell row blank equals cell square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus 6 squared end root end cell row blank equals cell square root of 9 plus 36 plus 36 end root end cell row blank equals cell square root of 81 end cell row blank equals 9 end table

Oleh karena itu, jawaban yang benar adalah D.

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