If $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} \times (\vec{a} \times \hat{i}) + \hat{j} \times (\vec{a} \times \hat{j}) + \hat{k} \times (\vec{a} \times \hat{k})$,then $|\vec{b}|$ is

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:

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 $\overline{a}, \overline{b}, \overline{c}$ be three non-zero vectors,such that no two of them are collinear and $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3}|\overline{b}||\overline{c}| \overline{a}$. If $\theta$ is the angle between the vectors $\overline{b}$ and $\overline{c}$,then the value of $\sin \theta$ is

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors having magnitudes $1, 1$ and $2$ respectively. If $\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0}$, then the acute angle between $\vec{a}$ and $\vec{c}$ is

Let $\overrightarrow{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}$,$\overrightarrow{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}$,and a vector $\vec{c}$ be such that $\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}$. If $\vec{a} \cdot \vec{c}=13$,then $(24-\vec{b} \cdot \vec{c})$ is equal to ...........