Two vectors acting in the opposite directions have a resultant of 10 units. If they act at right angles toeach other, then the resultant is 50 units. Calculate the magnitude of two vectors.

Options

A. `F_(1) =40 N, F_(2) =30 N`
B. `F_(1) =30 N, F_(2) =40 N`
C. `F_(1)=50 N, F_(2) =40 N`
D. `F_(1) = 100 N, F_(2) =60 N`

Answer

A

Explanation

Let `F_(1) and F_(2)` be the two forces where `F_(1) gt F_(2)`.
Then, `F_(1) -F_(2) =10 N" …(1)"`
`therefore F_(1)=F_(2) +10" ….(2)"`
But when `F_(1) bot F_(2)`,
`R= sqrt (F_(1)^(2)+F_(2)^(2))=50 N`
`therefore F_(1)^(2) + F_(2)^(2) =2500`
`therefore (F_(2) + 10)^(2) +F_(2)^(2) =2500" ....From (2)"`
`therefore F_(2)^(2) + 20 F_(2) +100 + F_(2)^(2) =2500`
`therefore 2F_(2)^(2) + 20 F_(2) =2500 -100 =2400`
`therefore F_(2)^(2) + 10F_(2) -1200 =0`
`therefore (F_(2) +40) (F_(2) -30) =0`
`therefore F_(2) =30 N` (Consider only the positive values)
`therefore F_(1) =30 + 10 =40 N`