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Pertanyaan

Diketahui vektor a = 4 i − 2 j ​ + 2 k dan vektor b = 2 i − 6 j ​ + 4 k . Proyeksi vektor ortogonal vektor a pada b adalah .... (UN 2011)

Diketahui vektor  dan vektor . Proyeksi vektor ortogonal vektor  pada  adalah ....(UN 2011)

  1. i with rightwards arrow on top minus j with rightwards arrow on top plus k with rightwards arrow on top

  2. i with rightwards arrow on top minus 3 j with rightwards arrow on top plus 2 k with rightwards arrow on top

  3. i with rightwards arrow on top minus 4 j with rightwards arrow on top plus 4 k with rightwards arrow on top

  4. 2 i with rightwards arrow on top minus j with rightwards arrow on top plus k with rightwards arrow on top

  5. 6 i with rightwards arrow on top minus 8 j with rightwards arrow on top plus 6 k with rightwards arrow on top

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Jawaban terverifikasi

Pembahasan

Diketahui vektor dan vektor Misal adalah proyeksi ortogonal vektor pada maka:

Diketahui vektor a with rightwards arrow on top equals 4 i with rightwards arrow on top minus 2 j with rightwards arrow on top plus 2 k with rightwards arrow on top dan vektor b with rightwards arrow on top equals 2 i with rightwards arrow on top minus 6 j with rightwards arrow on top plus 4 k with rightwards arrow on top

Misal

c with rightwards arrow on top adalah proyeksi ortogonal vektor a with rightwards arrow on top pada b with rightwards arrow on top maka:

c with rightwards arrow on top equals fraction numerator a with rightwards arrow on top. b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar squared end fraction. b with rightwards arrow on top

       equals fraction numerator 4.2 plus left parenthesis negative 2 right parenthesis. left parenthesis negative 6 right parenthesis plus 2.4 over denominator open parentheses square root of 2 squared plus left parenthesis negative 6 right parenthesis squared plus 4 squared end root close parentheses squared end fraction space left parenthesis 2 i with rightwards arrow on top minus 6 j with rightwards arrow on top plus 4 k with rightwards arrow on top right parenthesis

       equals 28 over open parentheses square root of 56 close parentheses squared space left parenthesis 2 i with rightwards arrow on top minus 6 j with rightwards arrow on top plus 4 k with rightwards arrow on top right parenthesis

       equals 28 over 56 space left parenthesis 2 i with rightwards arrow on top minus 6 j with rightwards arrow on top plus 4 k with rightwards arrow on top right parenthesis

       equals 1 half space left parenthesis 2 i with rightwards arrow on top minus 6 j with rightwards arrow on top plus 4 k with rightwards arrow on top right parenthesis

       equals i with rightwards arrow on top minus 3 j with rightwards arrow on top plus 2 k with rightwards arrow on top

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Diketahui vektor a = 5 i + 6 j ​ + k dan b = i − 2 j ​ − 2 k . Proyeksi ortogonal vektor a pada b adalah .... (UN 2012)

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