If $a = i + j - k$,$b = i - j + k$,and $c = i - j - k$,then $a \times (b \times c)$ is:

If $a, b$ and $c$ are three vectors with magnitudes $1, 1$ and $2$ respectively and $a \times (a \times c) + b = 0$, then the angle between $a$ and $c$ is

For unit vectors $\bar{a}, \bar{b}, \bar{c}$,if $\bar{a} \times (\bar{b} \times \bar{c}) = \frac{\bar{b}}{2}$ and $\bar{b}, \bar{c}$ are non-collinear vectors,then the angles made by $\bar{a}$ with $\bar{b}$ and $\bar{c}$ respectively are:

If $\vec{A} = \hat{i} - 2\hat{j} - 3\hat{k}$,$\vec{B} = 2\hat{i} + \hat{j} - \hat{k}$,and $\vec{C} = \hat{i} + 3\hat{j} - 2\hat{k}$,then $(\vec{A} \times \vec{B}) \times \vec{C} = \dots$

If $(\bar{a} \times \bar{b}) \times \bar{c} = -5 \bar{a} + 4 \bar{b}$ and $\bar{a} \cdot \bar{b} = 3$,then the value of $\bar{a} \times (\bar{b} \times \bar{c})$ is

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = \hat{i} + \hat{j}$,$\vec{c} = \hat{i}$ and $(\vec{a} \times \vec{b}) \times \vec{c} = \lambda \vec{a} + \mu \vec{b}$,then $\lambda + \mu$ is equal to :-