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

$\begin{array}{1 1} (A) \overrightarrow b\; is \;collinear \;vector. \\ (B) \overrightarrow b\; is \;unit \;vector. \\ (C) \overrightarrow a\; and \;\overrightarrow b\;are \;equal\; vectors. \\ (D) \overrightarrow b is\; any\; vector. \end{array}$

Toolbox:
• $\overrightarrow a.\overrightarrow b=0$ if $\overrightarrow a$ is perpendicular to $\overrightarrow b$
• $\overrightarrow a.\overrightarrow a=0$ if $\overrightarrow a=0$
Step 1:
$\overrightarrow a.\overrightarrow a=0$ and $\overrightarrow a.\overrightarrow b=0$
Then this implies if $\overrightarrow a=0$,then $\overrightarrow b$ is any vector.
Step 2:
Hence the conclusion is $\overrightarrow b$ is any vector.