# If $$\overrightarrow a.\overrightarrow a=0\: and \: \overrightarrow a.\overrightarrow b = 0$$, then what can be concluded about the vector $$\overrightarrow b$$?

Toolbox:
• $\overrightarrow a.\overrightarrow b=0$ if $\overrightarrow a$ is perpendicular to $\overrightarrow b$
• $\overrightarrow a.\overrightarrow a=0$ if $\overrightarrow a=0$
$\overrightarrow a.\overrightarrow a=0$ and $\overrightarrow a.\overrightarrow b=0$
Then this implies if $\overrightarrow a=0$,then $\overrightarrow b$ is any vector.
Hence the conclusion is $\overrightarrow b$ is any vector.