Question

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 \angle CAB,\angle ABC, \angle BCA

Right triangles and trigonometry
ANSWERED
asked 2021-06-07
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 \(\angle CAB,\angle ABC, \angle BCA\)

Expert Answers (1)

2021-06-08

\(AB=(5,-3,0)-(1,0,1)=(4,-3,1)\)
\(AC=(1,5,2)-(1,0,-1)=(0,5,3)\)
\(|AB|=\sqrt{4^2+3^2+1^2}=\sqrt{26}\)
\(|AC|=\sqrt{0+5^2+3^2}=\sqrt{34}\)
Use:
\(AB\cdot AC=|AB|\cdot|AC|\cos\angle CAB\)
\((4,-3,1)\cdot(0,5,3)=\sqrt{26}\cdot\sqrt{34}\cdot\cos\angle CAB\)
\(4\cdot0-3\cdot5+1\cdot3=\sqrt{26}\cdot\sqrt{34}\cdot\cos\angle CAB\)
\(-12=\sqrt{26}\cdot\sqrt{34}\cdot\cos\angle CAB\)
\(\cos\angle CAB=-0.404\)
\(\angle CAB=113.8\) degree
\(BA=(1,0,-1)-(5,-3,0)=(-4,3,-1)\)
\(BC=(1,5,2)-(5,-3,0)=(-4,8,2)\)
\(|BA|=\sqrt{26}\)
\(|BC|=\sqrt{4^2+8^2+2^2}=\sqrt{84}\)
Use:
\(BA\cdot BC=|BA|\cdot|BC|\cdot\cos\angle ABC\)
\((-4,3,-1)\cdot(-4,8,2)=\sqrt{26}\cdot\sqrt{84}\cdot\cos\angle ABC\)
\(16+24-2=\sqrt{26}\cdot\sqrt{84}\cdot\cos\angle ABC\)
\(\cos\angle ABC=0.813\)
\(\angle ABC=35.6\) degree
Sum of all \(\angle\) \(= 180\)
\(\angle BCA+35.6+113.8=180\)
\(\angle BCA=30.6\) degree

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