Which is greater, |$\vec a + \vec b$| or |$\vec a$|+|$\vec b$|? Pardon me if this has been asked before, I have encountered this question and its many variations in many physics mock papers.
I want to know if there is some sort of way to logically attempt these types of questions, I am not in requirement of an answer but a method which will help in solving questions of this form...
\begin{align*}
  |\mathbf{a}+\mathbf{b}|^{2} &amp;=
  (\mathbf{a}+\mathbf{b}) \cdot (\mathbf{a}+\mathbf{b}) \\
  &amp;= \mathbf{a} \cdot \mathbf{a}+\mathbf{a} \cdot \mathbf{b}+
     \mathbf{b} \cdot \mathbf{a}+\mathbf{b} \cdot \mathbf{b} \\
  &amp;= |\mathbf{a}|^{2}+2\, \mathbf{a} \cdot \mathbf{b}+|\mathbf{b}|^{2} \\
  &amp; \leq |\mathbf{a}|^{2}+2|\mathbf{a}||\mathbf{b}|+|\mathbf{b}|^{2}
         \quad (\because \: \mathbf{a} \cdot \mathbf{b} \leq
         |\mathbf{a}||\mathbf{b}|) \\
  &amp; = (|\mathbf{a}|+|\mathbf{b}|)^{2} \\
  \therefore \quad   |\mathbf{a}+\mathbf{b}|
  &amp; \leq |\mathbf{a}|+|\mathbf{b}|
         \quad (\because \: |\mathbf{a}+\mathbf{b}|, |\mathbf{a}|,
         |\mathbf{b}| \geq 0)
\end{align*}
|a|+|b| will have to be greater than or equal to |a+b| and we can see that without any math just using pure logic.
When you do a + b, b can do 2 things compliment a or destruct its value. Even in general math if a is positive and b is negative or vice versa, b would negate a, but on the contrary when you apply a || function first and then add, you are assuring that b always compliments a.
Hope it helps