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 ....

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

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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