Let $\vec{a}=-5 \hat{i}+\hat{j}-3 \hat{k}$,$\vec{b}=\hat{i}+2 \hat{j}-4 \hat{k}$ and $\vec{c}=(((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$. Then $\vec{c} \cdot(-\hat{i}+\hat{j}+\hat{k})$ is equal to

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = \alpha \vec{b} - \hat{n}, (\alpha \neq 0)$ and $\vec{b} \cdot \vec{c} = 12$,then $|\vec{c} \times (\vec{a} \times \vec{b})|$ is equal to:

If $(\bar{a} \times \bar{b}) \times \bar{c} = -5 \bar{a} + 4 \bar{b}$ and $\bar{a} \cdot \bar{b} = 3$,then the value of $\bar{a} \times (\bar{b} \times \bar{c})$ is

Let $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $\vec{c}=3 \hat{i}+2 \hat{j}-\hat{k}$ and its projection on $\vec{a}$ is $19 \text{ units}$,then $|2 \vec{v}|^{2}$ is equal to .... .

Let $\vec{a}$ be a non-zero vector. If $\vec{x}=\hat{i} \times(\vec{a} \times \hat{i})$, $\vec{y}=\hat{j} \times(\vec{a} \times \hat{j})-\vec{a}$ and $\vec{z}=\hat{k} \times(\vec{a} \times \hat{k})-\vec{a}$, then $\left[\begin{array}{lll}\vec{x} & \vec{y} & \vec{z}\end{array}\right]=$

If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(\overline{a}-2 \overline{b}) \cdot \{(\overline{a} \times \overline{b}) \times (2 \overline{a}+\overline{b})\}$ is