If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors,the angle between these vectors is ........ $^o$.

With respect to a rectangular cartesian coordinate system,three vectors are expressed as $\vec a = 4\hat i - \hat j$,$\vec b = -3\hat i + 2\hat j$,and $\vec c = -\hat k$,where $\hat i, \hat j, \hat k$ are unit vectors along the $X, Y,$ and $Z$-axis respectively. The unit vector $\hat r$ along the direction of the sum of these vectors is:

If $\vec{a}$ and $\vec{b}$ are two unit vectors inclined at an angle of $60^{\circ}$ to each other,then:

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is $\vec{R_1}$. If vector $\vec{B}$ is reversed,the resultant becomes $\vec{R_2}$. What is the value of $R_1^2 + R_2^2$?

If $|{\overrightarrow V _1} + {\overrightarrow V _2}| = |{\overrightarrow V _1} - {\overrightarrow V _2}|$ and ${V_2}$ is finite,then

Two forces $3\,N$ and $2\,N$ are at an angle $\theta$ such that the resultant is $R$. The first force is now increased to $6\,N$ and the resultant becomes $2R$. The value of $\theta$ is ....... $^\circ$.