 # 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 FobelloE 2021-06-07 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 $\mathrm{\angle }CAB,\mathrm{\angle }ABC,\mathrm{\angle }BCA$
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$AB=\left(5,-3,0\right)-\left(1,0,1\right)=\left(4,-3,1\right)$
$AC=\left(1,5,2\right)-\left(1,0,-1\right)=\left(0,5,3\right)$
$|AB|=\sqrt{{4}^{2}+{3}^{2}+{1}^{2}}=\sqrt{26}$
$|AC|=\sqrt{0+{5}^{2}+{3}^{2}}=\sqrt{34}$
Use:
$AB\cdot AC=|AB|\cdot |AC|\mathrm{cos}\mathrm{\angle }CAB$
$\left(4,-3,1\right)\cdot \left(0,5,3\right)=\sqrt{26}\cdot \sqrt{34}\cdot \mathrm{cos}\mathrm{\angle }CAB$
$4\cdot 0-3\cdot 5+1\cdot 3=\sqrt{26}\cdot \sqrt{34}\cdot \mathrm{cos}\mathrm{\angle }CAB$
$-12=\sqrt{26}\cdot \sqrt{34}\cdot \mathrm{cos}\mathrm{\angle }CAB$
$\mathrm{cos}\mathrm{\angle }CAB=-0.404$
$\mathrm{\angle }CAB=113.8$ degree
$BA=\left(1,0,-1\right)-\left(5,-3,0\right)=\left(-4,3,-1\right)$
$BC=\left(1,5,2\right)-\left(5,-3,0\right)=\left(-4,8,2\right)$
$|BA|=\sqrt{26}$
$|BC|=\sqrt{{4}^{2}+{8}^{2}+{2}^{2}}=\sqrt{84}$
Use:
$BA\cdot BC=|BA|\cdot |BC|\cdot \mathrm{cos}\mathrm{\angle }ABC$
$\left(-4,3,-1\right)\cdot \left(-4,8,2\right)=\sqrt{26}\cdot \sqrt{84}\cdot \mathrm{cos}\mathrm{\angle }ABC$
$16+24-2=\sqrt{26}\cdot \sqrt{84}\cdot \mathrm{cos}\mathrm{\angle }ABC$
$\mathrm{cos}\mathrm{\angle }ABC=0.813$
$\mathrm{\angle }ABC=35.6$ degree
Sum of all $\mathrm{\angle }$ $=180$
$\mathrm{\angle }BCA+35.6+113.8=180$
$\mathrm{\angle }BCA=30.6$ degree