The projection of $\bar{a} = \hat{i} - 2\hat{j} + \hat{k}$ on $\bar{b} = 2\hat{i} - \hat{j} + \hat{k}$ is

Let $\bar{a} = 2\bar{i} - \bar{j} + \bar{k}$,$\bar{b} = \bar{i} + 2\bar{j} - \bar{k}$,and $\bar{c} = \bar{i} + \bar{j} - 2\bar{k}$ be three vectors. $A$ vector $\bar{r}$ in the plane of $\bar{b}$ and $\bar{c}$ has a projection of magnitude $\sqrt{\frac{2}{3}}$ on the vector $\bar{a}$. Find $\bar{r}$.

If $e$ is a unit vector perpendicular to the plane determined by the points $2 \hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\hat{j}+\hat{k}$ and $-\hat{i}+\hat{j}-\hat{k}$. If $a=2 \hat{i}-3 \hat{j}+6 \hat{k}$, then the projection vector of $a$ on $e$ is

The projection of the vector $2i + j - 3k$ on the vector $i - 2j + k$ is:

The cosine of the angle between any two diagonals of a cube is

The shortest distance between the lines $r = 3i + 5j + 7k + \lambda(i + 2j + k)$ and $r = -i - j - k + \mu(7i - 6j + k)$ is