Question Number 58154 by behi83417@gmail.com last updated on 18/Apr/19
$$\boldsymbol{\mathrm{A}}\left(\mathrm{1},\mathrm{1}+\boldsymbol{\mathrm{i}}\right),\boldsymbol{\mathrm{B}}\left(\sqrt{\mathrm{2}}+\boldsymbol{\mathrm{i}},\mathrm{2}\right),\boldsymbol{\mathrm{C}}\left(\mathrm{1}−\mathrm{3}\boldsymbol{\mathrm{i}},\mathrm{1}−\boldsymbol{\mathrm{i}}\right) \\ $$$$\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{given}}. \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{angle}}\:\boldsymbol{\mathrm{between}}:\:\:\boldsymbol{\mathrm{AB}}\:\:\boldsymbol{\mathrm{and}}\:\:\boldsymbol{\mathrm{AC}}\:. \\ $$
Answered by MJS last updated on 18/Apr/19
$$\mathrm{if}\:\mathrm{i}=\sqrt{−\mathrm{1}}\:\mathrm{then}\:{A},\:{B},\:{C}\:\in\mathbb{R}^{\mathrm{4}} \:\mathrm{with}\:\mathrm{coordinates} \\ $$$${A}=\begin{pmatrix}{\mathrm{1}}\\{\mathrm{0}}\\{\mathrm{1}}\\{\mathrm{1}}\end{pmatrix}\:\:{B}=\begin{pmatrix}{\sqrt{\mathrm{2}}}\\{\mathrm{1}}\\{\mathrm{2}}\\{\mathrm{0}}\end{pmatrix}\:\:{C}=\begin{pmatrix}{\mathrm{1}}\\{−\mathrm{3}}\\{\mathrm{1}}\\{−\mathrm{1}}\end{pmatrix} \\ $$$$\mathrm{and}\:\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{how}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{angles}, \\ $$$$\mathrm{especially}\:\mathrm{of}\:\mathrm{skew}\:\mathrm{lines}\:\mathrm{in}\:\mathbb{R}^{\mathrm{4}} \\ $$
Commented by behi83417@gmail.com last updated on 18/Apr/19
$${dear}\:{MJS}\:{sir}!{it}\:{is}\:{not}\:{clear}\:{for}\:{me}. \\ $$$${please}\:{explain}\:{it}\:{bit}\:{more}! \\ $$
Commented by MJS last updated on 18/Apr/19
$${a}+{b}\mathrm{i}\:\in\mathbb{R}^{\mathrm{2}} \:\left(=\mathrm{the}\:\mathrm{complex}\:\mathrm{plane}\right) \\ $$$$\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{other}\:\mathrm{possibility}\:\mathrm{to}\:\mathrm{give}\:\mathrm{it}\:\mathrm{a}\:\mathrm{graphical} \\ $$$$\mathrm{equivalent}\:\mathrm{than}\:\mathrm{the}\:\mathrm{point}\:\begin{pmatrix}{{a}}\\{{b}}\end{pmatrix}\:\mathrm{with}\:{a},\:{b}\:\in\mathbb{R} \\ $$$$\Rightarrow\:\begin{pmatrix}{{a}+{b}\mathrm{i}}\\{{c}+{d}\mathrm{i}}\end{pmatrix}\:\mathrm{only}\:\mathrm{makes}\:\mathrm{sense}\:\mathrm{as}\:\begin{pmatrix}{{a}}\\{{b}}\\{{c}}\\{{d}}\end{pmatrix}\:\mathrm{with} \\ $$$${a},\:{b},\:{c},\:{d}\:\in\mathbb{R} \\ $$
Commented by behi83417@gmail.com last updated on 18/Apr/19
$${thanks}\:{in}\:{advance}\:{sir}\:{MJS}! \\ $$