Two forces of magnitudes $3 \, N$ and $4 \, N$ are acting on an object. If the angle between them is $90^\circ$,then their resultant force is ... $N$.

Given that $A_1+A_2=5 A_3$ and $A_1-A_2=3 A_3$,where $A_3=2 \hat{i}+4 \hat{j}$,find the value of $\frac{|A_1|}{|A_2|}$.

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 ....... $^o$

The unit vector parallel to the resultant of the vectors $\vec A = 4\hat i + 3\hat j + 6\hat k$ and $\vec B = - \hat i + 3\hat j - 8\hat k$ is

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

The resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ has a magnitude $R_{1}$. If the direction of $\overrightarrow{Q}$ is reversed,the resultant has a magnitude $R_{2}$. The value of $(R_{1}^{2} + R_{2}^{2})$ is: