Let $\vec{a} = \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$. If $\vec{a} \times \vec{b} + \vec{c} = \vec{0}$ and $\vec{a} \cdot \vec{b} = 3$,find the vector $\vec{b}$.

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b} \cdot \vec{c} = 0$ and $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} - \vec{c}}{2}$. If $\vec{d}$ is a vector such that $\vec{b} \cdot \vec{d} = \vec{a} \cdot \vec{b}$,then $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})$ is equal to

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

Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}$ and $\bar{c}=\hat{j}-\hat{k}$ be three vectors such that $\bar{a} \times \bar{b}=\bar{c}$ and $\bar{a} \cdot \bar{c}=0$. If the length of the projection vector of the vector $\bar{b}$ on the vector $\bar{a} \times \bar{c}$ is $l$,then the value of $3l^2$ is

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

If $\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k}$,$\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$,and $\vec{c} = \hat{i} + 3\hat{j} - \hat{k}$,then find $\vec{a} \times (\vec{b} \times \vec{c})$.