If the angle between bar{a} = xbar{i} - 3bar{ | vedclass

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(D) Given $\bar{a} = (x, -3, -1)$ and $\bar{b} = (2x, x, -1)$.$1$. The angle between $\bar{a}$ and $\bar{b}$ is acute,so $\bar{a} \cdot \bar{b} > 0$.$

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$x < 0$

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(D) Given $\bar{a} = (x, -3, -1)$ and $\bar{b} = (2x, x, -1)$.$1$. The angle between $\bar{a}$ and $\bar{b}$ is acute,so $\bar{a} \cdot \bar{b} > 0$.$(x)(2x) + (-3)(x) + (-1)(-1) > 0$$2x^2 - 3x + 1 > 0$$(2x - 1)(x - 1) > 0$This implies $x  1$. ... $(1)$$2$. The angle between $\bar{b}$ and the $y$-axis (vector $\bar{j} = (0, 1, 0)$) is obtuse,so $\bar{b} \cdot \bar{j} $(2x, x, -1) \cdot (0, 1, 0) $0 + x + 0 $x From $(1)$ and $(2)$,the intersection is $x < 0$.

If the angle between $\bar{a} = x\bar{i} - 3\bar{j} - \bar{k}$ and $\bar{b} = 2x\bar{i} + x\bar{j} - \bar{k}$ is acute and the angle between $\bar{b}$ and the $y$-axis is obtuse,then $x$ belongs to which interval?

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