Question

If $\overrightarrow{a}.\overrightarrow{a}=0\mathrm{and}\overrightarrow{a}.\overrightarrow{b}=0,$ what can you conclude about the vector $\overrightarrow{b}$?

Answer 1

$$\begin{gathered} \text{Given that}\overrightarrow{a}.\overrightarrow{a}=0 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2=0 \\ \Rightarrow \left|\overrightarrow{a}\right|=0...\left(1\right) \\ \text{Also, given that} \\ \overrightarrow{a}.\overrightarrow{b}=0 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =0\text{(where θ is the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}\text{)} \\ \Rightarrow 0\left|\overrightarrow{b}\right|\cos \theta =0[\text{From (1)]} \\ \Rightarrow 0=0 \\ \text{So, it means that for any vector}\overrightarrow{b}\text{, the given equation}\overrightarrow{a}.\overrightarrow{b}=0\text{is satisfied.} \end{gathered}$$