Five equal forces of $10 \, N$ each are applied at one point and all are lying in one plane. If the angles between them are equal,the resultant force will be ........... $N$.

The three vectors $\vec{A}=3 \hat{i}-2 \hat{j}+\hat{k}$,$\vec{B}=\hat{i}-3 \hat{j}+5 \hat{k}$ and $\vec{C}=2 \hat{i}-\hat{j}+4 \hat{k}$ will form

The maximum and minimum magnitude of the resultant of two given vectors are $17$ units and $7$ units respectively. If these two vectors are at right angles to each other,the magnitude of their resultant is

$A$ man travels $30 \ m$ along the direction of $3 \hat{i} + 4 \hat{j}$ and then moves '$d$' meters perpendicular to the initial direction such that his total displacement is along the $x$-axis. What is the value of '$d$' in meters?

If $\overrightarrow{ F }=2 \hat{ i }+\hat{ j }-\hat{ k }$ and $\overrightarrow{ r }=3 \hat{ i }+2 \hat{ j }-2 \hat{ k }$,then the scalar and vector products of $\overrightarrow{ F }$ and $\overrightarrow{ r }$ have the magnitudes respectively as

$A$ vector $\vec{A}$ having magnitude $6$ units is added to vector $\vec{B}$,which is along the $x$-axis. The resultant of $\vec{A}$ and $\vec{B}$ is along the $y$-axis. If the magnitude of the resultant of $\vec{A}$ and $\vec{B}$ is three times that of $\vec{B}$,then the magnitude of $\vec{B}$ is: