Find, correct to the nearest degree, the three angles of the triangle with the g

Yulia 2021-10-28 Answered
Find, correct to the nearest degree, the three angles of the triangle with the given vertices.
A(1,0,-1), B(5,-3,0), C(1,5,2)
Find \(\displaystyle\angle{C}{A}{B},\angle{A}{B}{C},\angle{B}{C}{A}\)

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Expert Answer

sovienesY
Answered 2021-10-29 Author has 9962 answers
\(\displaystyle{A}{B}={\left({5},-{3},{0}\right)}-{\left({1},{0},{1}\right)}={\left({4},-{3},{1}\right)}\)
\(\displaystyle{A}{C}={\left({1},{5},{2}\right)}-{\left({1},{0},-{1}\right)}={\left({0},{5},{3}\right)}\)
\(\displaystyle{\left|{A}{B}\right|}=\sqrt{{{4}^{{2}}+{3}^{{2}}+{1}^{{2}}}}=\sqrt{{{26}}}\)
\(\displaystyle{\left|{A}{C}\right|}=\sqrt{{{0}+{5}^{{2}}+{3}^{{2}}}}=\sqrt{{{34}}}\)
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
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