The resultant of two vectors at an angle $150^{\circ}$ is $10$ units and is perpendicular to one vector. The magnitude of the smaller vector is ....... units

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 $\frac{|A_1|}{|A_2|}$.

The $x$ and $y$ components of vector $\vec{P}$ have magnitudes $1$ and $3$,and the $x$ and $y$ components of the resultant of $\vec{P}$ and $\vec{Q}$ have magnitudes $5$ and $6$ respectively. What is the magnitude of $\vec{Q}$?

The resultant of two vectors $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. When the direction of $\vec{Q}$ is reversed,the resultant is given by $\vec{S}$. Which one of the following is true for vectors $\vec{R}$ and $\vec{S}$?

Which of the four arrangements in the figure correctly shows the vector addition of two forces $\overrightarrow {{F_1}} $ and $\overrightarrow {{F_2}} $ to yield the third force $\overrightarrow {{F_3}} $?

The resultant of two forces $3P$ and $2P$ is $R$. If the first force is doubled,then the resultant is also doubled. The angle between the two forces is ........... $^o$.