Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors such that $|\bar{a}|=1, |\bar{c}|=1, |\bar{b}|=4$,and $|\bar{b} \times \bar{c}|=\sqrt{15}$. If $\lambda \bar{a}=\bar{b}-2 \bar{c}$,then the value of $\lambda$ is

The point of intersection of $r \times a = b \times a$ and $r \times b = a \times b$,where $a = i + j$ and $b = 2i - k$ is

If the projection of $\bar{a}$ on $\bar{b}+\bar{c}$ is twice the projection of $\bar{b}+\bar{c}$ on $\bar{a}$,and if $|\bar{b}|=2 \sqrt{2}$,$|\bar{c}|=4$,and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{4}$,then $|\bar{a}|=$

If $\overline{a}+\overline{b}+\overline{c}=\overline{0}$ with $|\overline{a}|=3, |\overline{b}|=5$ and $|\overline{c}|=7$,then the angle between $\overline{a}$ and $\overline{b}$ is:

Let the point $A$ divide the line segment joining the points $P(-1, -1, 2)$ and $Q(5, 5, 10)$ internally in the ratio $r : 1$ $(r > 0)$. If $O$ is the origin and $(\overrightarrow{OQ} \cdot \overrightarrow{OA}) - \frac{1}{5}|\overrightarrow{OP} \times \overrightarrow{OA}|^2 = 10$,then the value of $r$ is:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a} + 2\vec{b} + 2\vec{c} = \vec{0}$,then $|\vec{a} \times \vec{c}|$ is equal to