The vertices of $\mathrm{△}\mathrm{ABC}$ are A(1, 4, 2), B(–2 , 1, 2), C(2, 3, –4). Find $\mathrm{\angle }\mathrm{A},\mathrm{\angle }\mathrm{B},\mathrm{\angle }\mathrm{C}$

Given points A = (1, 4, 2), B = (–2, 1, 2) C = (2, 3, –4)
$\text{ Dr's of }\overline{\mathrm{AB}}=\left({\mathrm{x}}_2-{\mathrm{x}}_1,{\mathrm{y}}_2-{\mathrm{y}}_1,{\mathrm{z}}_2-{\mathrm{z}}_1\right)$
$=(-2-1,1-4,2-2)=(-3,-3,0)$
$\text{ Dr's of }\overline{\mathrm{BC}}=\left({\mathrm{x}}_2-{\mathrm{x}}_1,{\mathrm{y}}_2-{\mathrm{y}}_1,{\mathrm{z}}_2-{\mathrm{z}}_1\right)$
$=(2+2,3-1,-4-2)=(4,2,-6)$
$\text{ Dr's of }\overline{\mathrm{CA}}=\left({\mathrm{x}}_2-{\mathrm{x}}_1,{\mathrm{y}}_2-{\mathrm{y}}_1,{\mathrm{z}}_2-{\mathrm{z}}_1\right)$
$=(1-2,4-3,2+4)=(-1,1,6)$
$\mathrm{A}\text{ is the angle between the lines }\overline{\mathrm{AB}}\text{ and }\overline{\mathrm{CA}}$
$\begin{array}{l}\cos \mathrm{A}=\frac{\left|{\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2+{\mathrm{c}}_1{\mathrm{c}}_2\right|}{\sqrt{{\mathrm{a}}_1^2+{\mathrm{b}}_1^2+{\mathrm{c}}_1^2}\sqrt{{\mathrm{a}}_2^2+{\mathrm{b}}_2^2+{\mathrm{c}}_2^2}}\\ \cos \mathrm{A}=\frac{\left|\right(-3\left)\right(-1)+(-3\left)\right(1)+0(6\left)\right|}{\sqrt{(-3{)}^2+(-3{)}^2+0^2}\sqrt{(-1{)}^2+1^2+6^2}}\\ =\frac{3-3}{\sqrt{18}\sqrt{38}}=0\Rightarrow \mathrm{\angle }\mathrm{A}={90}^0\end{array}$
$\mathrm{B}\text{ is the angle between the lines }\overline{\mathrm{AB}}\text{ and }\overline{\mathrm{BC}}$
$\begin{array}{l}\cos \mathrm{B}=\frac{\left|{\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2+{\mathrm{c}}_1{\mathrm{c}}_2\right|}{\sqrt{{\mathrm{a}}_1^2+{\mathrm{b}}_1^2+{\mathrm{c}}_1^2}\sqrt{{\mathrm{a}}_2^2+{\mathrm{b}}_2^2+{\mathrm{c}}_2^2}}\\ =\frac{|-3(4)-3(2)+0(-6\left)\right|}{\sqrt{9+9+0}\sqrt{16+4+36}}\\ =\frac{|-12-6|}{\sqrt{18}\sqrt{56}}\\ =\frac{18}{\sqrt{18}\sqrt{56}}\\ =\frac{\sqrt{18}}{\sqrt{56}}\\ =\sqrt{\frac{18}{56}}\\ =\sqrt{\frac{9}{28}}\\ =\frac{3}{2\sqrt{7}}\\ \Rightarrow \mathrm{B}={\cos }^{-1}\left(\frac{3}{2\sqrt{7}}\right)\end{array}$
$\mathrm{C}\text{ is the angle between the lines }\overline{\mathrm{CA}}\text{ and }\overline{\mathrm{BC}}$
$\begin{array}{l}\cos \mathrm{C}=\frac{\left|{\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2+{\mathrm{c}}_1{\mathrm{c}}_2\right|}{\sqrt{{\mathrm{a}}_1^2+{\mathrm{b}}_1^2+{\mathrm{c}}_1^2}\sqrt{{\mathrm{a}}_2^2+{\mathrm{b}}_2^2+{\mathrm{c}}_2^2}}\\ =\frac{|-1(4)+1(2)+6(-6\left)\right|}{\sqrt{1+1+36}\sqrt{16+4+36}}\\ =\frac{38}{\sqrt{38}\sqrt{56}}\\ =\frac{\sqrt{38}}{\sqrt{56}}\\ =\sqrt{\frac{38}{56}}\\ \cos \mathrm{C}=\sqrt{\frac{19}{28}}\Rightarrow \mathrm{C}={\cos }^{-1}\left(\sqrt{\frac{19}{28}}\right)\end{array}$