$\left|\overset{\rightarrow}{a}+\overset{\rightarrow}{b}\right|$ = $\left|\overset{\rightarrow}{a}\right|$ ⇒ $\left|\overset{\rightarrow}{a}+\overset{\rightarrow}{b}\right|^2$ = $\left|\overset{\rightarrow}{a}\right|^2$ ⇒ $\left|\overset{\rightarrow}{a}\right|^2$ + $\overset{\rightarrow}{2a}\cdot\overset{\rightarrow}{b}$ + $\left|\overset{\rightarrow}{b}\right|^2$ = $\left|\overset{\rightarrow}{a}\right|^2$ ⇒ $\overset{\rightarrow}{2a}\cdot\overset{\rightarrow}{b}$ + $\left|\overset{\rightarrow}{b}\right|^2$ = 0 ... (1) Now, $\overset{\rightarrow}{2a}\cdot\overset{\rightarrow}{b}$ . $\overset{\rightarrow}{b}$ = $\overset{\rightarrow}{2a}\cdot\overset{\rightarrow}{b}$ + $\overset{\rightarrow}{b}\cdot\overset{\rightarrow}{b}$ = $\overset{\rightarrow}{2a}\cdot\overset{\rightarrow}{b}$ + $\left|\overset{\rightarrow}{b}\right|^2$ = 0 We know that if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other. Thus,$\overset{\rightarrow}{2a}+\overset{\rightarrow}{b}$ is perpendicular to $\overset{\rightarrow}{b}$