Vectors 1 Question 27
28. Determine the value of $c$, so that for all real $x$, the vector $c x \hat{\mathbf{i}}-6 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $x \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 c x \hat{\mathbf{k}}$ make an obtuse angle with each other.
(1991, 4M)
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Answer:
Correct Answer: 28. (3)
Solution:
- Let $\overrightarrow{\mathbf{a}}=c x \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=x \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 c x \hat{\mathbf{k}}$. Since, $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ makes an obtuse angle.
$$ \begin{aligned} & \Rightarrow \quad \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}<0 \Rightarrow c x^{2}-12+6 c x<0 \\ & \Rightarrow \quad c<0 \text { and discriminant }<0 \\ & \Rightarrow \quad c<0 \quad \text { and } \quad 36 c^{2}-4 \cdot(-12) c<0 \\ & \Rightarrow \quad c<0 \quad \text { and } \quad 12 c(3 c+4)<0 \\ & \Rightarrow \quad c<0 \quad \text { and } \quad c>-4 / 3 \\ & \therefore \quad c \in(-4 / 3,0) \end{aligned} $$
