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

If $\vec{a} = 4\hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{c} = 3\hat{i} + 4\hat{j} + 5\hat{k}$, then $(\vec{a} + \vec{b}) \cdot (\vec{b} + \vec{c}) =$

If $|\vec{a}|=1, |\vec{b}|=2, |\vec{a}-\vec{b}|^2+|\vec{a}+2\vec{b}|^2=20$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is:

If $a$ and $b$ are two unit vectors such that $a+2b$ and $5a - 4b$ are perpendicular to each other,then the angle between $a$ and $b$ is ............. $^o$

The angle between the vectors $2 \hat{k} - 3 \hat{j}$ and $\hat{i} - 2 \hat{k}$ is

If $\overline{a}, \overline{b}, \overline{c}$ are three vectors such that $|\overline{a}+\overline{b}+\overline{c}|=1$,$\overline{c}=\lambda(\overline{a} \times \overline{b})$ and $|\overline{a}|=\frac{1}{\sqrt{3}}, |\overline{b}|=\frac{1}{\sqrt{2}}, |\overline{c}|=\frac{1}{\sqrt{6}}$,then the angle between $\bar{a}$ and $\bar{b}$ is