How can I know which angle will be given between two vectors using dot product

Question

$A,B,C$ have coordinates $(1,3,1),(2,7,-3),(4,-5,2)$. -Find angle ABC

Using the dot product method, I get $150.9$ but the answer is $29.1, (180-150.9)$ I understand that I will get the angle between the two vectors based on their direction, but how can I tell this is the case from the coordinates without having to draw a $3$D graph?

Answer 1

$A-B=(-1,-4,4)$, magnitude is $\sqrt{33}$

$C-B=(2,-12,5)$, magnitude is $\sqrt{173}$

dot product is $66$

$\therefore\cos(\theta)=\dfrac{66}{\sqrt{33}\sqrt{173}}$

$\therefore\theta\approx29.1^\circ$

Answer 2

Angle ABC means angle between $\vec{BA}$ and $\vec{BC}$.

$\vec{BA} = \vec{OA} - \vec{OB}$ and $\vec{BC} = \vec{OC} - \vec{OB}$

$$\vec{BA} . \vec{BC} = 66 = ||\vec{BA}|| ||\vec{BC}|| \cos(\theta) = \sqrt{33} \sqrt{173} \cos(\theta)$$

$\implies $ $\cos(\theta) = \frac{66}{\sqrt{33} \sqrt{173}}$