The resultant of two vectors $$\vec{A}$$ and $$\vec{B}$$ is perpendicular to $$\vec{A}$$ and its magnitude is half that of $$\vec{B}$$. The angle between vectors $$\vec{A}$$ and $$\vec{B}$$ is _________$$^\circ$$.

If $$\vec{a}$$ and $$\vec{b}$$ makes an angle $$\cos ^{-1}\left(\frac{5}{9}\right)$$ with each other, then $$|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|$$ for $$|\vec{a}|=n|\vec{b}|$$ The integer value of $$\mathrm{n}$$ is _________.

A vector has magnitude same as that of $$\vec{A}=3 \hat{i}+4 \hat{j}$$ and is parallel to $$\vec{B}=4 \hat{i}+3 \hat{j}$$. The $$x$$ and $$y$$ components of this vector in first quadrant are $$x$$ and 3 respectively where $$x=$$ _________.

If $$\overrightarrow P = 3\widehat i + \sqrt 3 \widehat j + 2\widehat k$$ and $$\overrightarrow Q = 4\widehat i + \sqrt 3 \widehat j + 2.5\widehat k$$ then, the unit vector in the direction of $$\overrightarrow P \times \overrightarrow Q $$ is $${1 \over x}\left( {\sqrt 3 \widehat i + \widehat j - 2\sqrt 3 \widehat k} \right)$$. The value of $$x$$ is _________.