What is the angle between $\overrightarrow P$ and the resultant of $(\overrightarrow P + \overrightarrow Q)$ and $(\overrightarrow P - \overrightarrow Q)$?

Which vector must be added to the resultant vector $\vec{P} = 2\hat{i} + 7\hat{j} - 10\hat{k}$ and $\vec{Q} = \hat{i} + 2\hat{j} + 3\hat{k}$ to obtain a unit vector along the $X$-axis?

The resultant of two vectors $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. If $\vec{Q}$ is doubled,the new resultant vector is perpendicular to $\vec{P}$. What is the magnitude of $\vec{R}$?

The magnitude of the $X$ and $Y$ components of $\vec{A}$ are $7$ and $6$ respectively. Also,the magnitude of the $X$ and $Y$ components of $\vec{A} + \vec{B}$ are $11$ and $9$ respectively. What is the magnitude of $\vec{B}$?

The angle between two forces of equal magnitude $R$,if the magnitude of their resultant is $\frac{R}{2}$,is:

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 ... $N$.