In a triangle $ABC$,if $2\overrightarrow{AC} = 3\overrightarrow{CB}$,then $2\overrightarrow{OA} + 3\overrightarrow{OB}$ equals

The projection of $\hat{i}-\hat{j}$ on $\hat{i}+\hat{j}$ is . . . . . . .

Vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are of the same length and they make equal angles with each other when taken in pairs. If $\vec{a} = \hat{i} - \hat{j}$,$\vec{b} = \hat{j} + \hat{k}$,and $\vec{c}$ makes an obtuse angle with the $x$-axis,find the vector $\vec{c}$.

If $\hat{x}, \hat{y},$ and $\hat{z}$ are three unit vectors in three-dimensional space,then find the minimum value of $|\hat{x} + \hat{y}|^2 + |\hat{y} + \hat{z}|^2 + |\hat{z} + \hat{x}|^2$.

If $\vec{r}_1 = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{r}_2 = \hat{i} + 2\hat{j} + 3\hat{k}$,find the unit vector parallel to the resultant vector of $\vec{r}_1$ and $\vec{r}_2$.

The position vector of point $C$ relative to $B$ is $(\hat{i} + \hat{j})$ and the position vector of $B$ relative to $A$ is $(\hat{i} - \hat{j})$. The position vector of $C$ relative to $A$ is: