If $\overline{a} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$ and $\overline{b} = \hat{i} - \hat{j} - \hat{k}$,then the projection of $\overline{b}$ in the direction of $\overline{a}$ is

$\bar{a}, \bar{b}, \bar{c}$ are three non-coplanar and mutually perpendicular vectors of same magnitude $K$. If $\bar{r}$ is any vector satisfying $\bar{a} \times ((\bar{r}-\bar{b}) \times \bar{a}) + \bar{b} \times ((\bar{r}-\bar{c}) \times \bar{b}) + \bar{c} \times ((\bar{r}-\bar{a}) \times \bar{c}) = \bar{0}$, then $\bar{r} =$

If $\hat{a}, \hat{b}$ and $\hat{c}$ are unit vectors such that $\hat{a}+\hat{b}+\hat{c}=\vec{0}$,then the value of $\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{c}+\hat{c} \cdot \hat{a}$ is equal to

If vectors satisfy the condition $|a - c| = |b - c|$,then $(b - a) \cdot \left( c - \frac{a + b}{2} \right)$ is equal to

If $|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=144$ and $|\vec{a}|=4$,then the value of $|\vec{b}|$ is

If $\vec{a}, \vec{b}, \text{ and } \vec{c}$ are non-coplanar vectors,then the point of intersection of the line passing through the points $2 \vec{a}+3 \vec{b}-\vec{c}$ and $3 \vec{a}+4 \vec{b}-2 \vec{c}$ with the line joining the points $\vec{a}-2 \vec{b}+3 \vec{c}$ and $\vec{a}-6 \vec{b}+6 \vec{c}$ is