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

Two forces $P + Q$ and $P - Q$ make an angle $2 \alpha$ with each other,and their resultant makes an angle $\theta$ with the bisector of the angle between them. Then:

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 $\vec{F}_1$ and $\vec{F}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{F}_1$ and $\vec{F}_2$ is $\cos^{-1}\left(\frac{1}{n}\right)$. The value of $|n|$ is . . . . . . .

What is the minimum number of vectors of equal magnitude required to produce a zero resultant vector?

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: