The sum of the lengths of projections of $p \hat{i} + q \hat{j} + r \hat{k}$ on the coordinate axes, where $p = 4, q = -5, r = 7$, is: (in $\text{ units}$)

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

If the magnitude of the sum of two unit vectors is greater than the magnitude of their difference and less than $\sqrt{3}$ times the magnitude of their difference,then the complete set of values for the angle $\theta$ between the vectors is

Three non-zero non-collinear vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $\vec{a}+3\vec{b}$ is collinear with $\vec{c}$, and $3\vec{b}+2\vec{c}$ is collinear with $\vec{a}$. Then $\vec{a}+3\vec{b}+2\vec{c}$ equals to

If $a = (2, 5)$ and $b = (1, 4),$ then the vector parallel to $(a + b)$ is

If the angle between $\vec{a}$ and $\vec{b}$ is $30^o$,then the angle between $3\vec{a}$ and $-4\vec{b}$ will be ............ $^o$.