The sum of the magnitudes of two forces acting at a point is 16 N. The resultant of these forces is perpendicular to the smaller force which has a magnitude of 8 N. If the magnitude of smaller force is x, then the value of x is

Explanation

Magnitude of the smaller vectors is denoted as x , and let the magnitude of other be y .
`x+y=16`
Let `theta` is the angle between two vetors . If resultant makes an nagle `alpha` with the smaller vector of magnitude x, then we can write the following :
`tan alpha=(y sin theta)/(x +y cos theta)`
Here `alpha =90^(@)`
`rArr tan 90^(@)=(y sin theta)/(x+y cos theta)= infty`
`rArr x+y cos theta=0`
Magnityde of the resultant of the two is given to be 8, hence we can write the following equation:
`x^(2)+y^(2)+2xy cos theta=(8)^(2)`
In the above equation we can substitute values from equations (i) and (ii) to get the following:
`x^(2)+(16-x)^(2)+2x(-x)=64`
`rArrx^(2)+256+x^(2)-32x-2x^(2)=64`
`rArr 32x=192 rArr x=6`