Question:
If $\vec{a} \cdot \vec{a}=0$ and $\vec{a} \cdot \vec{b}=0$, then what can be concluded about the vector $\vec{b} ?$
Solution:
It is given that $\vec{a} \cdot \vec{a}=0$ and $\vec{a} \cdot \vec{b}=0$.
Now,
$\vec{a} \cdot \vec{a}=0 \Rightarrow|\vec{a}|^{2}=0 \Rightarrow|\vec{a}|=0$
$\therefore \vec{a}$ is a zero vector.
Hence, vector $\vec{b}$ satisfying $\vec{a} \cdot \vec{b}=0$ can be any vector.