For the given vector $\vec{A} = 3\hat{i} - 4\hat{j} + 10\hat{k}$,the ratio of the magnitude of its component on the $x-y$ plane to the component on the $z$-axis is:

Consider a frame made up of two thin massless rods $AB$ and $AC$ as shown in the figure. $A$ vertical force $\overrightarrow{P}$ of magnitude $100 \; N$ is applied at point $A$ of the frame. Suppose the force $\overrightarrow{P}$ is resolved into components parallel to the arms $AB$ and $AC$ of the frame. The magnitude of the resolved component along the arm $AC$ is $x \; N$. The value of $x$,to the nearest integer,is ............

The component of a vector $\vec{r}$ along the $x$-axis will have a maximum value if

$A$ force of $5 \ N$ acts on a particle along a direction making an angle of $60^{\circ}$ with the vertical. Its vertical component will be $......... \ N$.

$A$ vector $\overrightarrow{A}$ makes equal angles with the $x$,$y$,and $z$ axes. Find the magnitude of its components.

If two forces of $5 \, N$ each are acting along $X$ and $Y$ axes,then the magnitude and direction of the resultant is