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Question Number 58154 by behi83417@gmail.com last updated on 18/Apr/19

$$\mathbf{A}(1,1+\mathbf{i}),\mathbf{B}(\sqrt{2}+\mathbf{i},2),\mathbf{C}(1-3\mathbf{i},1-\mathbf{i})$$$$\mathbf{are}\mathbf{given}.$$$$\mathbf{find}\mathbf{angle}\mathbf{between}:\mathbf{AB}\mathbf{and}\mathbf{AC}.$$

Answered by MJS last updated on 18/Apr/19

$$\mathrm{if}\mathrm{i}=\sqrt{-1}\mathrm{then}A,B,C\in {\mathbb{R}}^4\mathrm{with}\mathrm{coordinates}$$$$A=\left(\begin{array}{c}1\\ 0\\ 1\\ 1\end{array}\right)B=\left(\begin{array}{c}\sqrt{2}\\ 1\\ 2\\ 0\end{array}\right)C=\left(\begin{array}{c}1\\ -3\\ 1\\ -1\end{array}\right)$$$$\mathrm{and}\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}\mathrm{how}\mathrm{to}\mathrm{calculate}\mathrm{angles},$$$$\mathrm{especially}\mathrm{of}\mathrm{skew}\mathrm{lines}\mathrm{in}{\mathbb{R}}^4$$

Commented by behi83417@gmail.com last updated on 18/Apr/19

$$dearMJSsir!itisnotclearforme.$$$$pleaseexplainitbitmore!$$

Commented by MJS last updated on 18/Apr/19

$$a+b\mathrm{i}\in {\mathbb{R}}^2(=\mathrm{the}\mathrm{complex}\mathrm{plane})$$$${\mathrm{there}}^{\prime }\mathrm{s}\mathrm{no}\mathrm{other}\mathrm{possibility}\mathrm{to}\mathrm{give}\mathrm{it}\mathrm{a}\mathrm{graphical}$$$$\mathrm{equivalent}\mathrm{than}\mathrm{the}\mathrm{point}\left(\begin{array}{c}a\\ b\end{array}\right)\mathrm{with}a,b\in \mathbb{R}$$$$\Rightarrow \left(\begin{array}{c}a+b\mathrm{i}\\ c+d\mathrm{i}\end{array}\right)\mathrm{only}\mathrm{makes}\mathrm{sense}\mathrm{as}\left(\begin{array}{c}a\\ b\\ c\\ d\end{array}\right)\mathrm{with}$$$$a,b,c,d\in \mathbb{R}$$

Commented by behi83417@gmail.com last updated on 18/Apr/19

$$thanksinadvancesirMJS!$$

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