Why does triangle law of vector addition seem to disobey triangle inequality? I recently started learning about vectors,and one of the first things that was taught was Triangle law of vector addition. However I can't understand why triangle law of vector addition doesn't follow triangle inequality. 
For example, in the above diagram, shouldn't $R>A+B$ rather than being equal? Why is it not following triangle inequality?
 A: Note that the sides of this triangle have lengths equal to the magnitudes of the vectors $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{R}$, respectively.
And, the triangle inequality says that the length of the side representing $\mathbf{R}$ in particular cannot exceed the sum of the lengths of the sides representing $\mathbf{A}$ and $\mathbf{B}$. That is,
$$
\lvert \mathbf{R} \rvert \leq \lvert \mathbf{A} \rvert + \lvert \mathbf{B} \rvert, 
$$
or in other words,
$$
\lvert \mathbf{A} + \mathbf{B} \rvert \leq \lvert \mathbf{A} \rvert + \lvert \mathbf{B} \rvert. 
$$
A: The triangle inequality (i.e. sum of any two sides is always greater than the third one) holds true for lengths of sides in a triangle.
Therefore, in terms of lengths of vectors $|\vec A|, |\vec B|$ and $|\vec R|$ representing the sides of a triangle, the triangle inequality is satisfied as follows
$$|\vec R|<|\vec A|+|\vec B|$$
or $$R<A+B$$
Where, $R, A, $ and $B$ are the magnitudes of vectors $\vec R, \vec A$ and $\vec B$ respectively.
While the equality (triangle law) holds true for vector addition
$$\vec R=\vec A+\vec B$$
