Diketahui vektor \( \vec{a}= (1,2, -3), \ \vec{b} = (4,4,m) \) dan \( \vec{c} = (3,-4,5) \). Jika \( \vec{a} \) tegak lurus \( \vec{b} \), hasil dari \( \vec{a}+\vec{b}-2\vec{c} = \cdots \) (UN 2014)
- \( ( -1, \ 14, \ -9 ) \)
- \( ( -1, \ 14, \ -4 ) \)
- \( ( -1, \ 14, \ -3 ) \)
- \( ( -1, \ 14, \ -2 ) \)
- \( ( -1, \ 14, \ -1 ) \)
Pembahasan:
Jika vektor \( \vec{a} \) tegak lurus vektor \( \vec{b} \), maka
\begin{aligned} \vec{a} \cdot \vec{b} &= 0 \\[8pt] \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix} \cdot \begin{pmatrix} 4 \\ 4 \\ m \end{pmatrix} &= 0 \\[8pt] (1)(4)+(2)(4)+(-3)(m) &= 0 \\[8pt] 4+8-3m &=0 \\[8pt] 3m &= 12 \\[8pt] m &= 4 \\[8pt] \vec{b} &= \begin{pmatrix} 4 \\ 4 \\ 4 \end{pmatrix} \end{aligned}
Dengan demikian, hasil dari \( \vec{a}+\vec{b}-2\vec{c} \), yaitu:
\begin{aligned} \vec{a}+\vec{b}-2\vec{c} &= \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}+ \begin{pmatrix} 4 \\ 4 \\ 4 \end{pmatrix} -2 \begin{pmatrix} 3 \\ -4 \\ 5 \end{pmatrix} \\[8pt] &= \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}+ \begin{pmatrix} 4 \\ 4 \\ 4 \end{pmatrix} - \begin{pmatrix} 6 \\ -8 \\ 10 \end{pmatrix} \\[8pt] &= \begin{pmatrix} 1+4-6 \\ 2+4+8 \\ -3+4-10 \end{pmatrix} = \begin{pmatrix} -1 \\ 14 \\ -9 \end{pmatrix} \end{aligned}
Jawaban A.