1. Diketahui vektor \(\vec{a} = \begin{pmatrix}2 \\ 1 \\ -1 \end{pmatrix}\), \(\vec{b} = \begin{pmatrix}3 \\ 3 \\ m \end{pmatrix}\), dan \(\vec{c} = \begin{pmatrix}1 \\ -2 \\ 4 \end{pmatrix}\). Jika \(\vec{a} \perp \vec{b}\), maka \(\vec{a} \cdot \vec{b} = 0\).
2. Hitung dot product:
$$
\vec{a} \cdot \vec{b} = 2 \times 3 + 1 \times 3 + (-1) \times m = 6 + 3 - m = 9 - m
$$
Karena \(\vec{a} \perp \vec{b}\), maka:
$$
9 - m = 0 \implies m = 9
$$
3. Substitusi nilai \(m = 9\) ke \(\vec{b}\):
$$
\vec{b} = \begin{pmatrix}3 \\ 3 \\ 9 \end{pmatrix}
$$
4. Hitung \(\vec{a} + 2\vec{b} - \vec{c}\):
$$
\vec{a} + 2\vec{b} - \vec{c} = \begin{pmatrix}2 \\ 1 \\ -1 \end{pmatrix} + 2 \times \begin{pmatrix}3 \\ 3 \\ 9 \end{pmatrix} - \begin{pmatrix}1 \\ -2 \\ 4 \end{pmatrix} = \begin{pmatrix}2 \\ 1 \\ -1 \end{pmatrix} + \begin{pmatrix}6 \\ 6 \\ 18 \end{pmatrix} - \begin{pmatrix}1 \\ -2 \\ 4 \end{pmatrix}
$$
5. Jumlahkan dan kurangkan komponen vektor:
$$
= \begin{pmatrix}2 + 6 - 1 \\ 1 + 6 - (-2) \\ -1 + 18 - 4 \end{pmatrix} = \begin{pmatrix}7 \\ 9 \\ 13 \end{pmatrix}
$$
Jadi, hasil \(\vec{a} + 2\vec{b} - \vec{c} = \begin{pmatrix}7 \\ 9 \\ 13 \end{pmatrix}\).
Vektor Orthogonal 59Bb0C
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