If the vectors overline{a} = c(log_7 x) hat{i | vedclass

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(C) Let $y = \log_7 x$. Since $x > 0$,$y$ can take any real value in $(-\infty, \infty)$.The vectors are $\overline{a} = (cy) \hat{i} + 2 \hat{j} + 3

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$\left(-\frac{4}{3}, 0\right)$

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(C) Let $y = \log_7 x$. Since $x > 0$,$y$ can take any real value in $(-\infty, \infty)$.The vectors are $\overline{a} = (cy) \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $\overline{b} = y \hat{i} + (3cy) \hat{j} - 4 \hat{k}$.For the vectors to make an obtuse angle,their dot product must be negative: $\overline{a} \cdot \overline{b} $\overline{a} \cdot \overline{b} = (cy)(y) + (2)(3cy) + (3)(-4) $cy^2 + 6cy - 12 For this quadratic expression in $y$ to be negative for all $y \in \mathbb{R}$,the coefficient of $y^2$ must be negative $(c $D = (6c)^2 - 4(c)(-12) $36c^2 + 48c $12c(3c + 4) The roots are $c = 0$ and $c = -4/3$. The inequality holds for $c \in (-4/3, 0)$.Since we also require $c < 0$,the intersection is $c \in (-4/3, 0)$.

If the vectors $\overline{a} = c(\log_7 x) \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $\overline{b} = (\log_7 x) \hat{i} + 3c(\log_7 x) \hat{j} - 4 \hat{k}$ make an obtuse angle for any $x > 0$,then $c$ belongs to

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