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Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k},\) \(\vec{b}=\hat{i}\) and \(\vec{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}\) then

(a) Let \(c_1=1\) and \(c_2=2,\) find \(c_3\) which makes \(\vec{a}, \vec{b}\) and \(\vec{c}\) coplanar.

(b) If \(c_2=-1\) and \(c_3=1,\) show that no value of \(c_1\) can make \(\vec{a}, \vec{b}\) and \(\vec{c}\) coplanar.

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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.

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