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

The correct options are
A $40,30$
D $30,40$
Let $\overrightarrow{P}\&\overrightarrow{Q}$ are two vectors.
Case 1: $P>Q$
For opposite direction,
$P-Q=10$ unit -------(1)
For perpendicular direction,
$\sqrt{P^2+Q^2}=50$ unit
$P^2+Q^2={50}^2.....\left(2\right)$
From equation (1) and (2)
$(10+Q{)}^2+Q^2={50}^2$
$2Q^2+20Q+100=2500$
$2Q^2+20Q=2400$
$Q^2+10Q-1200=0$
$(Q+40)(Q-30)=0$
$Q=30$ units (we cannot take negative sign because magnitude cannot be negative)
So, from equation (1)
$P=10+Q$
$=10+30$
$P=40$ units

Case 2: $P<Q$
Simillarly, from equation 1 and 2
$(Q-10{)}^2+Q^2={50}^2$
Then, $Q=40$ Units
Then, $P=Q-10$
$P=30$ Units