Question

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

Answer 1

${\displaystyle \overrightarrow{a}\times \overrightarrow{a}=0,\overrightarrow{a}\times \overrightarrow{b}=0}$

${\displaystyle \overrightarrow{a}\times \overrightarrow{a}=0}$

${\displaystyle {\left|\overrightarrow{a}\right|}^2=0}$

${\displaystyle \left|\overrightarrow{a}\right|=0,\overrightarrow{a}\times \overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =0}$

So ${\displaystyle \overrightarrow{b}}$ can be any vectors.