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
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 $$\angle CAB,\angle ABC, \angle BCA$$

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