Discussion
If the angle between the vectors \(\vec{a} = x\hat{i} + 3\hat{j} + \hat{k}\) and \(\vec{b} = x\hat{i} - x\hat{j} + 2\hat{k}\) is acute, then \(x\) lies is
Answer: C
x lies in \((-\infty, 1) \cup (2, \infty)\).
The angle between two vectors is acute when their dot product is positive.
\[ \begin{aligned} \vec{a}\cdot\vec{b} &= x^{2} - 3x + 2 \\ &= (x - 1)(x - 2) \gt 0 \end{aligned} \]
The product is positive when both factors have the same sign: \(x \lt 1\) or \(x \gt 2\). The other options use the wrong roots, such as -1 or 0, which come from sign slips in \(-3x\).
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