# If $B$ is added to $A$, under what condition does the resultant vector have a magnitude equal to $A+B$?

If $B$ is added to $A$, under what condition does the resultant vector have a magnitude equal to $A+B$? Under what conditions is the resultant vector equal to zero?

My Attempt: Let $\theta$ be the angle between $\vec {A}$ and $\vec {B}$. Then,

$$R=A+B$$ $$\sqrt {A^2+B^2+2A.B\cos \theta}=A+B$$ $$A^2+B^2+2A.B\cos \theta=A^2+B^2+2A.B$$ $$\cos \theta =1$$ $$\cos \theta = \cos 0$$ $$\theta =0°$$ The resultant have a magnitude $A+B$ when the angle between the vectors is $0°$. How do I solve the second part of the question.?

• First point looks good. For the second one, all you need is $\,A+B=0 \iff A = -B\,$.
– dxiv
Sep 13, 2017 at 2:31
• @dxiv, And how do I get that? Please elaborate.
– pi-π
Sep 13, 2017 at 2:39
• First $|A|=|B|$, then $\cos\theta=-1$ Sep 13, 2017 at 2:46
• @blue_eyed_... how do I get that? It's part of the definition of a vector space that every vector $V$ has an additive inverse $-V$ such that $V + (-V) = 0\,$. Then, if $A+B=0$ you just add $(-B)$ to both sides of the equality and get $A=-B\,$.
– dxiv
Sep 13, 2017 at 3:18
• Technically the answer to the question is "always," because if $A$ and $B$ are vectors then $A+B$ is the resultant of adding $B$ to $A$ and of course it has the same magnitude as itself. Presumably the person who asked the question meant to ask when the magnitude of $A+B$ is equal to the sum of the magnitude of $A$ and the magnitude of $B.$ Aug 21, 2021 at 2:25

Though the obvious fact that $\vec B=-\vec A$, we can infer this from your previous expression:
$$A^2+B^2+2AB\cos\theta=0$$
If $A=0$, then trivially $B^2=0 \to B=0$, and viceversa, and the angle can be any value.
If $A,B\gt0$, this leads to: $$\cos\theta=-{A^2+B^2 \over 2AB}$$
Knowing that $\cos(\cdot)$ must be bounded in [-1,1]: $$-1\le {A^2+B^2 \over 2AB}\le1\\ -2AB\le A^2+B^2\le2AB\\ -4AB\le (A-B)^2\le0\\$$ which can only be true for $A=B$, and the angle in this case is $\cos\theta=-{2\over2}=-1 \to \theta=\pi$