If for two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$,the sum $(\overrightarrow{A} + \overrightarrow{B})$ is perpendicular to the difference $(\overrightarrow{A} - \overrightarrow{B})$,then the ratio of their magnitudes is:

Two vectors $\vec{A}$ and $\vec{B}$ are defined as $\vec{A} = a \hat{i}$ and $\vec{B} = a(\cos \omega t \hat{i} + \sin \omega t \hat{j})$,where $a$ is a constant and $\omega = \pi / 6 \text{ rad s}^{-1}$. If $|\vec{A} + \vec{B}| = \sqrt{3}|\vec{A} - \vec{B}|$ at time $t = \tau$ for the first time,the value of $\tau$,in seconds,is . . . . . .

Match Column-$I$ with Column-$II$.

Column-$I$	Column-$II$
$(1)$ Resultant of two vectors is maximum	$(a)$ $180^o$
$(2)$ Resultant of two vectors is minimum	$(b)$ $90^o$
	$(c)$ $0^o$

Two forces $P$ and $Q$,of magnitude $2F$ and $3F$,respectively,are at an angle $\theta$ with each other. If the force $Q$ is doubled,then their resultant also gets doubled. Then,the angle $\theta$ is ....... $^o$

If $\sqrt{A^2+B^2}$ represents the magnitude of the resultant of two vectors $(\vec{A}+\vec{B})$ and $(\vec{A}-\vec{B})$,then the angle between the two vectors $\vec{A}$ and $\vec{B}$ is

$A$ unit vector in the direction of the resultant vector of $\vec{A} = -2 \hat{i} + 3 \hat{j} + \hat{k}$ and $\vec{B} = \hat{i} + 2 \hat{j} - 4 \hat{k}$ is