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

If $a = 3i - j + 2k,$ $b = 2i + j - k$ and $c = i - 2j + 2k,$ then $(a \times b) \times c$ is equal to

Let $a, b, c$ be three vectors. Determine the correctness of the following statements:
$(i)$ $(a \times b) \times c = (a \cdot c) b - (b \cdot c) a$
(ii) $a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$

Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k}$,$\vec{a} \cdot \vec{b}=1$ and $\vec{a} \times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6 \vec{b}$ is equal to

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b} \times(\vec{b} \times \vec{c}) .$ If magnitudes of the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are $\sqrt{2}, 1$ and $2$ respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta$ $(0 < \theta < \frac{\pi}{2})$,then the value of $1+\tan \theta$ is equal to:

Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b} + \vec{c})$. If $\vec{b}$ is not parallel to $\vec{c}$,then the angle between $\vec{a}$ and $\vec{b}$ is: