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 $^{\circ}$)

Let $\vec{a}=-\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=\hat{i}-\hat{j}-3\hat{k}$, $\vec{c}=\vec{a}\times\vec{b}$ and $\vec{d}=\vec{c}\times\vec{a}$. Then $(\vec{a}-\vec{b}) \cdot \vec{d}$ is equal to :

If $(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$ where $\vec{a}, \vec{b},$ and $\vec{c}$ are any three vectors such that $\vec{a} \cdot \vec{b} \neq 0$ and $\vec{b} \cdot \vec{c} \neq 0$,then $\vec{a}$ and $\vec{c}$ are:

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:

The vectors $\bar{a}$ and $\bar{b}$ are not perpendicular and $\overline{c}$ and $\overline{d}$ are two vectors satisfying $\overline{b} \times \overline{c} = \overline{b} \times \overline{d}$ and $\overline{a} \cdot \overline{d} = 0$. Then the vector $\overline{d}$ is equal to:

If $a = i + j - 2k$, then $\sum \{(a \times i) \times j\}^2$ is equal to