Answer:
To find the magnitudes of the forces P and Q, we can use vector addition and some trigonometry. Let's denote:
P = magnitude of force P
Q = magnitude of force Q
R = magnitude of the resultant force R = 180 N
We are given that the sum of forces P and Q is 270 N, so we can write:
P + Q = 270 N
We are also given that the angle between force P and the resultant R is 90°. When two vectors are at a right angle to each other, we can use the Pythagorean theorem to relate their magnitudes.
So, we have:
P^2 + R^2 = Q^2
Now, we know the values of R and the equation relating P and Q. Let's substitute R = 180 N into the equation:
P^2 + (180 N)^2 = Q^2
Now, we can use the equation P + Q = 270 N to solve for one of the variables. Let's solve for P:
P = 270 N - Q
Now, substitute this expression for P into the equation involving P and Q:
(270 N - Q)^2 + (180 N)^2 = Q^2
Now, we can solve this equation for Q. After finding Q, we can use P + Q = 270 N to find the value of P.
I'll calculate these values:
(270 N - Q)^2 + (180 N)^2 = Q^2
Expanding and simplifying:
72900 N^2 - 540 NQ + Q^2 + 32400 N^2 = Q^2
Combine like terms:
105300 N^2 - 540 NQ + 32400 N^2 = Q^2
137700 N^2 - 540 NQ = Q^2
Now, we can rearrange the equation:
Q^2 - 540 NQ - 137700 N^2 = 0
We can solve this quadratic equation for Q using the quadratic formula:
Q = [540 N ± sqrt((540 N)^2 - 4(137700 N^2))]/(2)
Q = [540 N ± sqrt(291600 N^2 - 550800 N^2)]/(2)
Q = [540 N ± sqrt(-259200 N^2)]/(2)
Since the term inside the square root is negative, there are no real solutions for Q. This means there might be an issue with the given information or setup of the problem.
Please double-check the problem statement or provide additional information if necessary.