For any four points $A$, $B$, $C$ and $D$, which of the following statements are true?

I. $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$.

II. $\overrightarrow{AB} - \overrightarrow{AC} = \overrightarrow{CB}$.

III. $\overrightarrow{AB} + \overrightarrow{BA} = 2\overrightarrow{AB}$.

IV. $\lvert \overrightarrow{AB} + \overrightarrow{BD} \rvert = \lvert \overrightarrow{AB} \rvert + \lvert \overrightarrow{BD} \rvert$.

Explanation

1. 01

   I is the triangle law. II: $\overrightarrow{AB} - \overrightarrow{AC} = (\vec{b} - \vec{a}) - (\vec{c} - \vec{a}) = \vec{b} - \vec{c} = \overrightarrow{CB}$.

2. 02

   III is false: $\overrightarrow{BA} = -\overrightarrow{AB}$, so the sum is $\vec{0}$. IV is false unless $B$ lies on the segment $AD$: the left side is $\lvert \overrightarrow{AD} \rvert$, a side of triangle $ABD$, which is shorter than the other two sides together.