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Diketahui A(4, 0, 0), B(0, -4, 0), dan C(0, 0, 4)....

Diketahui A(4, 0, 0), B(0, -4, 0), dan C(0, 0, 4). 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 ....

  1. begin mathsize 14px style fraction numerator 3 square root of 2 over denominator 2 end fraction end style

  2. begin mathsize 14px style fraction numerator square root of 2 over denominator 2 end fraction end style

  3. begin mathsize 14px style fraction numerator square root of 2 over denominator 3 end fraction end style

  4. begin mathsize 14px style 2 square root of 2 end style

  5. begin mathsize 14px style fraction numerator square root of 3 over denominator 2 end fraction end style

Jawaban:

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

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