For all real \(x\), the vectors \(C x \hat{i}-6 \hat{j}-3 \hat{k}\) and \(x \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{C} x \hat{\mathrm{k}}\) make an obtuse angle with each other, then the value of C can be in

Let \(\overline{\mathrm{a}}=C x \hat{\mathrm{i}}-6 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}\) and
\(\overline{\mathrm{b}}=x \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{C} x \hat{\mathrm{k}}\)
Angle between \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) is obtuse.
\(\therefore \overline{\mathrm{a}} \cdot \overline{\mathrm{~b}} \lt \cos 180^{\circ} \)
\( \therefore \overline{\mathrm{a}} \cdot \overline{\mathrm{~b}} \lt 0 \)
\( \therefore \mathrm{C} x^2-12-6 \mathrm{C} x \lt 0\)
\(\Rightarrow \mathrm{C} x^2-6 \mathrm{C} x-12 \lt 0 \)
\( \Rightarrow \mathrm{C} \lt 0 \text { and } \mathrm{D} \lt 0 \)
\( \Rightarrow \mathrm{C} \lt 0 \text { and } 36 \mathrm{C}^2+48 \mathrm{C} \lt 0 \)
\( \Rightarrow \mathrm{C} \lt 0 \text { and } 3 \mathrm{C}^2+4 \mathrm{C} \lt 0 \)
\( \Rightarrow \mathrm{C} \lt 0, \mathrm{C}(3 \mathrm{C}+4) \lt 0 \)
\( \Rightarrow \mathrm{C} \lt 0,-\frac{4}{3} \lt \mathrm{C} \lt 0 \)
\( \therefore \mathrm{C} =\left(-\frac{4}{3}, 0\right)\)