If the vectors $\bar{a}=\hat{i}-\hat{j}+2 \hat{k}$,$\bar{b}=2 \hat{i}+4 \hat{j}+\hat{k}$,and $\bar{c}=p \hat{i}+\hat{j}+q \hat{k}$ are mutually orthogonal,then $(p, q)$ is equal to

Let the unit vectors $a$ and $b$ be perpendicular and the unit vector $c$ be inclined at an angle $\theta$ to both $a$ and $b$. If $c = \alpha a + \beta b + \gamma (a \times b)$,then

Let $\vec{a}, \vec{b}, \vec{c}$ be non-coplanar vectors. If the three points with position vectors $\lambda \vec{a}-2 \vec{b}+\vec{c}$, $2 \vec{a}+\lambda \vec{b}-2 \vec{c}$, and $4 \vec{a}+7 \vec{b}-8 \vec{c}$ are collinear, then $\lambda=$

Let $ABC$ be a triangle such that $\overrightarrow{BC} = \overrightarrow{a}$,$\overrightarrow{CA} = \overrightarrow{b}$,$\overrightarrow{AB} = \overrightarrow{c}$,$|\overrightarrow{a}| = 6\sqrt{2}$,$|\overrightarrow{b}| = 2\sqrt{3}$,and $\overrightarrow{b} \cdot \overrightarrow{c} = 12$. Consider the statements:
$(S1): |(\overrightarrow{a} \times \overrightarrow{b}) + (\overrightarrow{c} \times \overrightarrow{b})| - |\overrightarrow{c}| = 6(2\sqrt{2} - 1)$
$(S2): \angle ABC = \cos^{-1}\left(\sqrt{\frac{2}{3}}\right)$.
Which of the following is true?

If $(2, -1, 2)$ and $(K, 3, 5)$ are the triads of direction ratios of two lines and the angle between them is $45^{\circ}$,then the value of $K$ is

If $\frac{\pi}{2} < \theta \leq \pi$ and $|\overline{a}|=5, |\overline{b}|=13, |\overline{a} \times \overline{b}|=25$,then the value of $\overline{a} \cdot \overline{b}$ is