Let F1 and F2 be two forces.
CASE 1:
when forces are acting in opposite direction then angle between F1and F2 is 180
so F1+F2=10
=> F1^2 + F2^2 +2F1F2cos 180=100
=>F1^2 + F2^2 -2F1F2=100
=>F1-F2=10

CASE 2:
when forces are acting perpendicularly then angle between F1and F2 is 90
so F1+F2=50
=>F1^2+F2^2+2 F1F2cos90=50^2
=>F1+F2=2500
=>(10+F2)^2+F2^2=2500
=>F2=30

hence F2=30 N
F1=40 N

Problem: If two forces act in opposite direction then their resultant is 10N and if they act mutually perpendicular their resultant is 50N. Find the magnitude of both the forces.

Solution:

To solve this problem, we can use the Pythagorean theorem to calculate the magnitude of each force.

Step 1: Determine the magnitude of the first force.

Let's assume that the magnitude of the first force is x N.

If two forces act in opposite directions, their resultant is given by the difference between the magnitudes of the two forces. In this case, we know that the resultant is 10N, so we can write:

x - y = 10 (where y is the magnitude of the second force)

Step 2: Determine the magnitude of the second force.

If two forces act mutually perpendicular to each other, their resultant is given by the square root of the sum of the squares of the two forces. In this case, we know that the resultant is 50N, so we can write:

sqrt(x^2 + y^2) = 50

Squaring both sides, we get:

x^2 + y^2 = 2500

Step 3: Solve the system of equations.

Now we have two equations and two unknowns, so we can solve for x and y. Rearranging the first equation, we get:

y = x - 10

Substituting this into the second equation, we get:

x^2 + (x - 10)^2 = 2500

Expanding, we get:

2x^2 - 20x + 100 = 2500

Simplifying, we get:

x^2 - 10x + 700 = 0

Using the quadratic formula, we get:

x = 10 ± sqrt(100 - 2800)/2

x = 10 ± sqrt(2700)/2

x = 10 ± 15

Since we can't have a negative magnitude, we discard the negative solution and get:

x = 25 N

Substituting this into the first equation, we get:

y = 25 - 10

y = 15 N

Therefore, the magnitude of the first force is 25 N and the magnitude of the second force is 15 N.

Conclusion: The magnitude of the two forces are 25 N and 15 N respectively.