If $\overrightarrow x$ and $\overrightarrow y$ be two non-zero vectors such that $$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$ and ${2\overrightarrow x + \lambda \overrightarrow y }$ is perpendicular to ${\overrightarrow y }$, then the value of $\lambda$ is _________ .

$$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$ <br>Squaring both sides we get <br><br>$${\left| {\overrightarrow x } \right|^2} + 2\overrightarrow x .\overrightarrow y + {\left| {\overrightarrow y } \right|^2} = {\left| {\overrightarrow x } \right|^2}$$ <br><br>$\Rightarrow$ $2\overrightarrow x .\overrightarrow y + \overrightarrow y .\overrightarrow y$ = 0 ....(1) <br><br>Given ${2\overrightarrow x + \lambda \overrightarrow y }$ is perpendicular to ${\overrightarrow y }$ <br><br>$\therefore$ $$\left( {2\overrightarrow x + \lambda \overrightarrow y } \right).\overrightarrow y $$ = 0 <br><br>$\Rightarrow$ $$2\overrightarrow x .\overrightarrow y + \lambda \overrightarrow y .\overrightarrow y $$ = 0 ....(2) <br><br>Comparing (1) &amp; (2) we get, $\lambda$ = 1