Given \(\vec{a}=\hat{i}+\hat{j}+\hat{k};\) \(\vec{b}=\hat{i}\) and \(\vec{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}\)
(a) Since \(\vec{a}\,\vec{b}\) and \(\vec{c}\) vectors are coplanar
(b) To make \(\vec{a}\,\vec{b}\) and \(\vec{c}\) coplanar.
\(\Rightarrow -1-1=0\)
\(\Rightarrow -2=0\) which is never possible.
Hence, if \(c_2=-1\) and \(c_3=1,\) there is no value of \(c_1\) which can make \(\vec{a}\,\,\vec{b}\) and \(\vec{c}\) coplanar.