Question

If the resultant of the two forces is of magnitude P and equal to one of them and perpendicular to it , the the other force is

Answer 1

Let the other force be F and resolve with respect to the axes.

Let P, the original force be along the x axis, so it makes an angle 0 and it can be written as P with direction x. (Or i if you prefer that).

Let the force F make an angle $\theta$

For the resultant to be perpendicular, it must have no component in the x direction.
Which means the vertical components of the original forces cancel,

$Fcos\theta=-P$

So the only component of resultant is given by $Fsin\theta$

This means $Fsin\theta=P$

Dividing these equations we get,

$cot\theta=-1$
$\theta=135$

Substituting this value in earlier equations,

$Fsin(135)=P$
$F=P \sqrt{2}$