Question

If the angle between the vectors $x\hat{i}+3\hat{j}-7\hat{k}\mathrm{and}x\hat{i}-x\hat{j}+4\hat{k}$ is acute, then x lies in the interval
(a) (−4, 7)
(b) [−4, 7]
(c) R −[−4, 7]
(d) R −(4, 7)

Answer 1

(c) R −[−4, 7]

$$\begin{gathered} \text{Let θ be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}\text{.} \\ \cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}=\frac{x^2-3x-28}{\sqrt{x^2+3^2+49}\sqrt{x^2+x^2+4^2}} \\ \text{For θ to be acute,} \\ \cos \theta >0 \\ \Rightarrow x^2-3x-28>0 \\ \Rightarrow \left(x-7\right)\left(x+4\right)>0 \\ \Rightarrow x\in \left(-\infty ,-4\right)\cup \left(7,\infty \right) \\ \Rightarrow x\in R-\left[-4,7\right] \end{gathered}$$