Number of unit vectors of the form a i ^ + b j ^ + c k ^ , where a ,   b ,   c ∈ W is

If $a\hat{\mathrm{i}}+b\hat{\mathrm{j}}+c\hat{\mathrm{k}}$ is an unit vector, then $\sqrt{a^2+b^2+c^2}=1$

$\Rightarrow a^2+b^2+c^2=1$

If $a, b, c\in W$, then possible values of $\left(a,b,c\right)$ are $\left(1,0,0\right), \left(0,1,0\right), \left(0,0,1\right)$

$\therefore$ Three unit vectors in the form of $a\hat{\mathrm{i}}+b\hat{\mathrm{j}}+c\hat{\mathrm{k}},$where $a, b, c\in W$