[연습문제] 벡터연산
\(\begin{split}&\overrightarrow{a}=a_1\hat{e}_1+a_2\hat{e}_2+a_3\hat{e}_3\\&\overrightarrow{b}=b_1\hat{e}_1+b_2\hat{e}_2+b_3\hat{e}_3\\&\overrightarrow{c}=c_1\hat{e}_1+c_2\hat{e}_2+c_3\hat{e}_3\end{split}\)
1. \(\overrightarrow{a}\cdot\overrightarrow{b}\) (dot product)
(a) \(\begin{Bmatrix}a_1&a_2&a_3\end{Bmatrix}\begin{Bmatrix}a_1\\a_2\\a_3\end{Bmatrix}=a_1b_1+a_2b_2+a_3b_3\)
(b) \(a_ib_i=a_1b_1+a_2b_2+a_3b_3\)
2. \(\overrightarrow{a}\times\overrightarrow{b}\) (cross product)
(a) \(\begin{Bmatrix}a_1&a_2&a_3\end{Bmatrix}\times\begin{Bmatrix}b_1&b_2&b_3\end{Bmatrix}=\left|\begin{matrix}\hat{e}_1&\hat{e}_2&\hat{e}_3\\a_1&a_2&a_3\\b_1&b_2&b_3\end{matrix}\right|\)
\(=(a_2b_3-a_3b_2)\hat{e}_1+(a_3b_1-a_1b_3)\hat{e}_2+(a_1b_2-a_2b_1)\hat{e}_3\)
(b) \(a_i\times b_j=\epsilon_{ijk}a_jb_k\)
\(=(\epsilon_{123}a_2b_3+\epsilon_{132}a_3b_2)\hat{e}_1+(\epsilon_{131}a_3b_1+\epsilon_{213}a_1b_3)\hat{e}_2+(\epsilon_{312}a_1b_2+\epsilon_{321}a_2b_1)\hat{e}_3\)
\(=(a_2b_3-a_3b_2)\hat{e}_1+(a_3b_1-a_1b_3)\hat{e}_2+(a_1b_2-a_2b_1)\hat{e}_3\)
3. \(\overrightarrow{a}\otimes\overrightarrow{b}\) (tensor product)
(a) \(\begin{Bmatrix}a_1&a_2&a_3\end{Bmatrix}\otimes\begin{Bmatrix}b_1&b_2&b_3\end{Bmatrix}=\begin{bmatrix}a_1b_1&a_1b_2&a_1b_3\\a_2b_1&a_2b_2&a_2b_3\\a_3b_1&a_3b_2&a_3b_3\end{bmatrix}\)
(b) \(a_ib_j=\begin{bmatrix}a_1b_1&a_1b_2&a_1b_3\\a_2b_1&a_2b_2&a_2b_3\\a_3b_1&a_3b_2&a_3b_3\end{bmatrix}\)
4. \(\overrightarrow{a}\cdot\left(\overrightarrow{b}\times\overrightarrow{c}\right)\)
(a) \(\begin{Bmatrix}a_1&a_2&a_3\end{Bmatrix}\left|\begin{matrix}\hat{e}_1&\hat{e}_2&\hat{e}_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{matrix}\right|=\begin{Bmatrix}a_1&a_2&a_3\end{Bmatrix}\begin{Bmatrix}b_2c_3-b_3c_2\\b_3c_1-b_1c_3\\b_1c_2-b_2c_1\end{Bmatrix}\)
\(=a_1b_2c_3-a_1b_3c_2-a_2b_1c_3+a_2b_3c_1+a_3b_1c_2-a_3b_2c_1\)
(b) \(a_i\cdot\left(\epsilon_{ijk}b_jc_k\right)=\epsilon_{ijk}a_ib_jc_k\)
\(=\epsilon_{123}a_1b_2c_3+\epsilon_{132}a_1b_3c_2+\epsilon_{213}a_2b_1c_3+\epsilon_{231}a_2b_3c_1+\epsilon_{312}a_3b_1c_2+\epsilon_{321}a_3b_2c_1\)
\(=a_1b_2c_3-a_1b_3c_2-a_2b_1c_3+a_2b_3c_1+a_3b_1c_2-a_3b_2c_1\)
5. \(\overrightarrow{a}\times\left(\overrightarrow{b}\times\overrightarrow{c}\right)\)
(a) \(\begin{Bmatrix}a_1&a_2&a_3\end{Bmatrix}\times\left|\begin{matrix}\hat{e}_1&\hat{e}_2&\hat{e}_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{matrix}\right|=\begin{Bmatrix}a_1&a_2&a_3\end{Bmatrix}\times\begin{Bmatrix}b_2c_3-b_3c_2\\b_3c_1-b_1c_3\\b_1c_2-b_2c_1\end{Bmatrix}\)
\(=\left|\begin{matrix}\hat{e}_1&\hat{e}_2&\hat{e}_3\\a_1&a_2&a_3\\b_2c_3-b_3c_2&b_3c_1-b_1c_3&b_1c_2-b_2c_1\end{matrix}\right|=(a_2b_1c_2-a_2b_2c_1-a_3b_3c_1+a_3b_1c_3)\hat{e}_1\)
\(+(a_3b_2c_3-a_3b_3c_2-a_1b_1c_2+a_1b_2c_1)\hat{e}_2+(a_1b_3c_1-a_1b_1c_3-a_2b_2c_3+a_2b_3c_2)\hat{e}_3\)
(b) \(a_i\times(\epsilon_{lkm}b_lc_m)=\epsilon_{ijk}\epsilon_{klm}a_jb_lc_m\)
\(=(\epsilon_{123}\epsilon_{312}a_2b_1c_2+\epsilon_{123}\epsilon_{321}a_2b_2c_1+\epsilon_{132}\epsilon_{231}a_3b_3c_1+\epsilon_{132}\epsilon_{213}a_3b_1c_3)\hat{e}_1\)
\(+(\epsilon_{231}\epsilon_{123}a_3b_2c_3+\epsilon_{231}\epsilon_{132}a_3b_3c_2+\epsilon_{213}\epsilon_{312}a_1b_1c_2+\epsilon_{213}\epsilon_{321}a_1b_2c_1)\hat{e}_2\)
\(+(\epsilon_{312}\epsilon_{231}a_1b_3c_1+\epsilon_{312}\epsilon_{213}a_1b_1c_3+\epsilon_{321}\epsilon_{123}a_2b_2c_3+\epsilon_{321}\epsilon_{132}a_2b_3c_2)\hat{e}_3\)
\(=(a_2b_1c_2-a_2b_2c_1-a_3b_3c_1+a_3b_1c_3)\hat{e}_1\)
\(+(a_3b_2c_3-a_3b_3c_2-a_1b_1c_2+a_1b_2c_1)\hat{e}_2\)
\(+(a_1b_3c_1-a_1b_1c_3-a_2b_2c_3+a_2b_3c_2)\hat{e}_3\)
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