The ratio of the maximum and minimum magnitudes of the resultant of two vectors $\vec{a}$ and $\vec{b}$ is $3 : 1$. Then $|\vec{a}|$ is equal to:

The sum of the magnitudes of two vectors acting at a point is $18$ and the magnitude of their resultant is $12$. If the resultant is at $90^{\circ}$ with the vector of smaller magnitude,then the magnitudes of the vectors are

Resultant of two vectors $\vec{P}$ and $\vec{Q}$ is of magnitude $A$. If $\vec{Q}$ is reversed, then the resultant is of magnitude $B$. The value of $A^2 + B^2$ 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

If three vectors have equal magnitude,i.e.,$A = B = C$,then the angle between $\vec{A}$ and $\vec{C}$ is $\alpha$. If $\vec{A} + \vec{B} + \vec{C} = 0$,then the angle between $\vec{A}$ and $\vec{C}$ is $\beta$. Find the ratio $\frac{\alpha}{\beta}$.

The sum of the magnitudes of two vectors $\vec{A}$ and $\vec{B}$ is $8$,and the magnitude of their resultant is $4$. If the resultant vector is perpendicular to one of the vectors,then the magnitudes of the two vectors $\vec{A}$ and $\vec{B}$ are: