Let $a, b, c$ be three unit vectors such that $a \times(b \times c)=\frac{1}{2} b$. If the angle between $a$ and $b$ is $\theta_1$ and the angle between $a$ and $c$ is $\theta_2$, then $\theta_1+\theta_2$ is equal to (in $^{\circ}$)

$(\vec{a} \times \vec{b}) \times [(\vec{b} \times \vec{c}) \times (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})]$ is

If $\overline{a}, \overline{b}, \overline{c}$ are three vectors with magnitudes $\sqrt{3}, 1, 2$ respectively,such that $\overline{a} \times (\overline{a} \times \overline{c}) + 3 \overline{b} = \overline{0}$,and if $\theta$ is the angle between $\overline{a}$ and $\overline{c}$,then $\sec^2 \theta$ is:

If $a = i + j + k$,$b = i + j$,$c = i$ and $(a \times b) \times c = \lambda a + \mu b$,then $\lambda + \mu = \dots$

Let $\overline{a}, \overline{b},$ and $\overline{c}$ be non-zero vectors such that $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3} |\overline{b}| |\overline{c}| \overline{a}.$ If $\theta$ is the angle between $\overline{b}$ and $\overline{c},$ then $\sin \theta = .....$

Let $a, b$ and $c$ be three unit vectors such that $a \times (b \times c) = \frac{1}{\sqrt{2}}(b + c)$ and $b$ is not parallel to $c$. If $\alpha$ and $\beta$ are the angles between $a, b$ and $a, c$ respectively, then $\alpha - \beta =$