Luas Segitiga di Ruang $\mathbb{R}^n$

[cotrans]

Sanctus, Sanctus, Dominus Deus Sabaoth.

\section{Luas Segitiga di Ruang $\mathbb{R}^n$}

Luas segitiga di ruang $\mathbb{R}^n$ yang ketiga titik sudutnya $\vec{a} := \sum_{i = 1}^n a_i\hat{x}_i$, $\vec{b} := \sum_{i = 1}^n b_i\hat{x}_i$, dan $\vec{c} := \sum_{i = 1}^n c_i\hat{x}_i$ di mana $a_i, b_i, c_i \in \mathbb{R}$ untuk semua $i \in \{ 1, \cdots, n \}$, serta
\[ \hat{x}_i := (\underset{n}{\underbrace{0, \cdots, 0, \overset{i}{1}, 0, \cdots, 0}}), \]
adalah
\[ \Delta := \frac{1}{2}|\vec{a}\times\vec{b} + \vec{b}\times\vec{c} + \vec{c}\times\vec{a}|. \]
\[ \Delta = \frac{1}{2}\left|\sum_{i, j = 1}^n a_ib_j\hat{x}_i\times\hat{x}_j + \sum_{i, j = 1}^n b_ic_j\hat{x}_i\times\hat{x}_j + \sum_{i, j = 1}^n c_ia_j\hat{x}_i\times\hat{x}_j\right|. \]
\[ \Delta = \frac{1}{2}\left|\sum_{i, j = 1}^n (a_ib_j + b_ic_j + c_ia_j)\hat{x}_i\times\hat{x}_j\right|. \]
\[ \Delta = \frac{1}{2}\left[\sum_{i, j = 1}^n (a_ib_j + b_ic_j + c_ia_j)(\hat{x}_i\times\hat{x}_j)\cdot\sum_{k, l = 1}^n (a_kb_l + b_kc_l + c_ka_l)(\hat{x}_k\times\hat{x}_l)\right]^{1/2}. \]
\[ \Delta = \frac{1}{2}\left[\sum_{i, j, k , l = 1}^n (a_ib_j + b_ic_j + c_ia_j)(a_kb_l + b_kc_l + c_ka_l)(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})\right]^{1/2}. \]
\[ \Delta = \frac{1}{2}\left[\sum_{i, j = 1}^n (a_ib_j + b_ic_j + c_ia_j)[(a_ib_j + b_ic_j + c_ia_j) - (a_jb_i + b_jc_i + c_ja_i)]\right]^{1/2}. \]

Apabila $n = 1$, maka $\Delta = 0$.

Apabila $n = 2$, maka
\[ \Delta = \frac{1}{2}\left|\begin{vmatrix} a_1 & a_2 & 1 \\ b_1 & b_2 & 1 \\ c_1 & c_2 & 1 \end{vmatrix}\right|. \]

Apabila $n = 3$, maka
\[ \Delta = \frac{1}{2}\left[\begin{vmatrix} 1 & a_2 & a_3 \\ 1 & b_2 & b_3 \\ 1 & c_2 & c_3 \end{vmatrix}^2 + \begin{vmatrix} a_1 & 1 & a_3 \\ b_1 & 1 & b_3 \\ c_1 & 1 & c_3 \end{vmatrix}^2 + \begin{vmatrix} a_1 & a_2 & 1 \\ b_1 & b_2 & 1 \\ c_1 & c_2 & 1 \end{vmatrix}^2\right]^{1/2}. \]

Sampai jumpa lagi.