\(\displaystyle{A}{B}={\left({5},-{3},{0}\right)}-{\left({1},{0},{1}\right)}={\left({4},-{3},{1}\right)}\)

\(\displaystyle{A}{C}={\left({1},{5},{2}\right)}-{\left({1},{0},-{1}\right)}={\left({0},{5},{3}\right)}\)

\(\displaystyle{\left|{A}{B}\right|}=\sqrt{{{4}^{{2}}+{3}^{{2}}+{1}^{{2}}}}=\sqrt{{{26}}}\)

\(\displaystyle{\left|{A}{C}\right|}=\sqrt{{{0}+{5}^{{2}}+{3}^{{2}}}}=\sqrt{{{34}}}\)

Use:

\(\displaystyle{A}{B}\cdot{A}{C}={\left|{A}{B}\right|}\cdot{\left|{A}{C}\right|}{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle{\left({4},-{3},{1}\right)}\cdot{\left({0},{5},{3}\right)}=\sqrt{{{26}}}\cdot\sqrt{{{34}}}\cdot{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle{4}\cdot{0}-{3}\cdot{5}+{1}\cdot{3}=\sqrt{{{26}}}\cdot\sqrt{{{34}}}\cdot{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle-{12}=\sqrt{{{26}}}\cdot\sqrt{{{34}}}\cdot{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle{\cos{\angle}}{C}{A}{B}=-{0.404}\)

\(\displaystyle\angle{C}{A}{B}={113.8}\) degree

\(\displaystyle{B}{A}={\left({1},{0},-{1}\right)}-{\left({5},-{3},{0}\right)}={\left(-{4},{3},-{1}\right)}\)

\(\displaystyle{B}{C}={\left({1},{5},{2}\right)}-{\left({5},-{3},{0}\right)}={\left(-{4},{8},{2}\right)}\)

\(\displaystyle{\left|{B}{A}\right|}=\sqrt{{{26}}}\)

\(\displaystyle{\left|{B}{C}\right|}=\sqrt{{{4}^{{2}}+{8}^{{2}}+{2}^{{2}}}}=\sqrt{{{84}}}\)

Use:

\(\displaystyle{B}{A}\cdot{B}{C}={\left|{B}{A}\right|}\cdot{\left|{B}{C}\right|}\cdot{\cos{\angle}}{A}{B}{C}\)

\(\displaystyle{\left(-{4},{3},-{1}\right)}\cdot{\left(-{4},{8},{2}\right)}=\sqrt{{{26}}}\cdot\sqrt{{{84}}}\cdot{\cos{\angle}}{A}{B}{C}\)

\(\displaystyle{16}+{24}-{2}=\sqrt{{{26}}}\cdot\sqrt{{{84}}}\cdot{\cos{\angle}}{A}{B}{C}\)

\(\displaystyle{\cos{\angle}}{A}{B}{C}={0.813}\)

\(\displaystyle\angle{A}{B}{C}={35.6}\) degree

Sum of all angle = 180

\(\displaystyle\angle{B}{C}{A}+{35.6}+{113.8}={180}\)

\(\displaystyle\angle{B}{C}{A}={30.6}\) degree

\(\displaystyle{A}{C}={\left({1},{5},{2}\right)}-{\left({1},{0},-{1}\right)}={\left({0},{5},{3}\right)}\)

\(\displaystyle{\left|{A}{B}\right|}=\sqrt{{{4}^{{2}}+{3}^{{2}}+{1}^{{2}}}}=\sqrt{{{26}}}\)

\(\displaystyle{\left|{A}{C}\right|}=\sqrt{{{0}+{5}^{{2}}+{3}^{{2}}}}=\sqrt{{{34}}}\)

Use:

\(\displaystyle{A}{B}\cdot{A}{C}={\left|{A}{B}\right|}\cdot{\left|{A}{C}\right|}{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle{\left({4},-{3},{1}\right)}\cdot{\left({0},{5},{3}\right)}=\sqrt{{{26}}}\cdot\sqrt{{{34}}}\cdot{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle{4}\cdot{0}-{3}\cdot{5}+{1}\cdot{3}=\sqrt{{{26}}}\cdot\sqrt{{{34}}}\cdot{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle-{12}=\sqrt{{{26}}}\cdot\sqrt{{{34}}}\cdot{\cos{\angle}}{C}{A}{B}\)

\(\displaystyle{\cos{\angle}}{C}{A}{B}=-{0.404}\)

\(\displaystyle\angle{C}{A}{B}={113.8}\) degree

\(\displaystyle{B}{A}={\left({1},{0},-{1}\right)}-{\left({5},-{3},{0}\right)}={\left(-{4},{3},-{1}\right)}\)

\(\displaystyle{B}{C}={\left({1},{5},{2}\right)}-{\left({5},-{3},{0}\right)}={\left(-{4},{8},{2}\right)}\)

\(\displaystyle{\left|{B}{A}\right|}=\sqrt{{{26}}}\)

\(\displaystyle{\left|{B}{C}\right|}=\sqrt{{{4}^{{2}}+{8}^{{2}}+{2}^{{2}}}}=\sqrt{{{84}}}\)

Use:

\(\displaystyle{B}{A}\cdot{B}{C}={\left|{B}{A}\right|}\cdot{\left|{B}{C}\right|}\cdot{\cos{\angle}}{A}{B}{C}\)

\(\displaystyle{\left(-{4},{3},-{1}\right)}\cdot{\left(-{4},{8},{2}\right)}=\sqrt{{{26}}}\cdot\sqrt{{{84}}}\cdot{\cos{\angle}}{A}{B}{C}\)

\(\displaystyle{16}+{24}-{2}=\sqrt{{{26}}}\cdot\sqrt{{{84}}}\cdot{\cos{\angle}}{A}{B}{C}\)

\(\displaystyle{\cos{\angle}}{A}{B}{C}={0.813}\)

\(\displaystyle\angle{A}{B}{C}={35.6}\) degree

Sum of all angle = 180

\(\displaystyle\angle{B}{C}{A}+{35.6}+{113.8}={180}\)

\(\displaystyle\angle{B}{C}{A}={30.6}\) degree