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Question
If the angle between the vectors \[x \hat{i} + 3 \hat{j}- 7 \hat{k} \text{ and } x \hat{i} - x \hat{j} + 4 \hat{k}\] is acute, then x lies in the interval
Options
(a) (−4, 7)
(b) [−4, 7]
(c) R −[−4, 7]
(d) R −(4, 7)
Solution
(c) R −[−4, 7]
\[\text{ Let }\theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} .\]
\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{x^2 - 3x - 28}{\sqrt{x^2 + 3^2 + 49} \sqrt{x^2 + x^2 + 4^2}}\]
\[\text{ For }\theta \text{ to be acute },\]
\[\cos \theta > 0\]
\[ \Rightarrow x^2 - 3x - 28 > 0\]
\[ \Rightarrow \left( x - 7 \right)\left( x + 4 \right) > 0\]
\[ \Rightarrow x \in \left( - \infty , - 4 \right) \cup \left( 7, \infty \right)\]
\[ \Rightarrow x \in R - \left[ - 4, 7 \right]\]
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