Is the magnitude of the sum of two vectors equal to the sum of their magnitudes individually?

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If the sum of two vectors is zero then the magnitude of the two vectors is the same, however, if the sum of three or more vectors is zero then their magnitude may or may not be the same? Select one True or False

In $\mathbb{R}^n$, we have that your equality implies $$\langle a+b,a+b \rangle=(\Vert a\Vert+\Vert b \Vert)^2 ,$$ which, using the bilinearity of the inner product, yields $$\Vert a \Vert^2+\Vert b \Vert^2 +2\langle a,b \rangle=\Vert a\Vert^2+\Vert b \Vert ^2 +2\Vert a \Vert \Vert b \Vert$$ $$\implies \langle a,b \rangle= \Vert a \Vert \Vert b \Vert.$$ And equality on Cauchy-Schwarz only holds if both vectors are linearly dependent. (Note that since there is no modulus on the inner product on the left side, not only they must be linearly dependent, but also differ by a positive scaling).

As a sidenote, the technique above holds for any inner product space. One might be tempted to extend it to any normed space, but the result isn't true. As an example, one can take $L^1([0,1])$ and $a=I_{[0,1/2]}$, $b=I_{[1/2,1]}$, where $I_A$ is the indicator function on $A$.