Let $\vec{a}=3 \hat{i}+\hat{j}$ and $\vec{b}=\hat{i}+2 \hat{j}+\hat{k}$. Let $\vec{c}$ be a vector satisfying $\vec{a} \times(\vec{b} \times \vec{c})=\vec{b}+\lambda \vec{c}$. If $\vec{b}$ and $\vec{c}$ are non-parallel,then the value of $\lambda$ is.

If $a \cdot b = \beta$ and $a \times b = c$, then $a =$

If $a, b, c$ are non-coplanar unit vectors such that $a \times (b \times c) = \frac{b + c}{\sqrt{2}}$,then the angle between $a$ and $b$ is

Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$,$\vec{b}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c}=5$,then $|\vec{c}|$ is equal to

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar unit vectors such that $\bar{a} \times (\bar{b} \times \bar{c}) = \frac{\bar{b} + \bar{c}}{\sqrt{2}}$,then the angle between $\bar{a}$ and $\bar{b}$ is:

Let $\overline{a}, \overline{b}, \overline{c}$ be three non-zero vectors,such that no two of them are collinear and $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3}|\overline{b}||\overline{c}| \overline{a}$. If $\theta$ is the angle between the vectors $\overline{b}$ and $\overline{c}$,then the value of $\sin \theta$ is