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 ...........

$(b \times c) \times (c \times a) = \dots$

If $a=2 \hat{i}-3 \hat{j}+\hat{k}$, $b=\hat{i}-\hat{j}+2 \hat{k}$ and $c=2 \hat{i}+\hat{j}+\hat{k}$ are three vectors, then $|(a \times b) \times c|=$

Let $\overline{a}, \overline{b}$ and $\overline{c}$ 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 vectors $\overline{b}$ and $\overline{c}$,then the value of $\sin \theta$ is

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