The speed of a particle changes from $\sqrt{5} \ m/s$ to $2\sqrt{5} \ m/s$ in a time $t$. If the magnitude of change in its velocity is $5 \ m/s$,the angle between the initial and final velocities of the particle is (in $^{\circ}$)

The angle made by the resultant vector of two vectors $2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $2 \hat{i}-7 \hat{j}-4 \hat{k}$ with the $x$-axis is (in $^{\circ}$)

Among three vectors,two vectors are equal in magnitude,and the magnitude of the third vector is $\sqrt{2}$ times the magnitude of the other two vectors. If $\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} = 0$,find the angles between the vectors.

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	$(b)$ $90^o$
	$(c)$ $0^o$

If the resultant vector of vectors $\vec{A} = 4\hat{i} + 3\hat{j} + 6\hat{k}$ and $\vec{B} = -\hat{i} + 3\hat{j} - 8\hat{k}$ is parallel to a unit vector,then $\vec{R}$ is:

$A$ vector $\vec Q$ which has a magnitude of $8$ is added to the vector $\vec P$ which lies along the $x$-axis. The resultant of the two vectors lies along the $y$-axis and has a magnitude twice that of $\vec P$. The magnitude of $\vec P$ is: