The resultant of $\vec{A} + \vec{B}$ is $\vec{R}_1$. On reversing the vector $\vec{B}$,the resultant becomes $\vec{R}_2$. What is the value of $R_1^2 + R_2^2$?

The magnitudes of two vectors are $8$ units and $6$ units respectively. Find the magnitude of the resultant vector if the angle between these two vectors is $(i) \theta = 0^{\circ}$,$(ii) \theta = 180^{\circ}$,$(iii) \theta = 90^{\circ}$,and $(iv) \theta = 120^{\circ}$.

The magnitudes of two vectors are $A$ and $B$ $(A > B)$. If the maximum resultant magnitude of the two vectors is $n$ times their minimum resultant magnitude,then $\frac{A}{B} =$

The vector that must be added to the vectors $\hat{i} - 3\hat{j} + 2\hat{k}$ and $3\hat{i} + 6\hat{j} - 7\hat{k}$ so that the resultant vector is a unit vector along the $y$-axis is

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 $\vec{Q}$?

Two vectors $\vec{A}$ and $\vec{B}$ have equal magnitudes. Then the vector $\vec{A} + \vec{B}$ is perpendicular to: