Final answer:
The problem is about finding the ratio of magnitudes of two vectors P and Q under different inclinations. The resultant vector is derived using vector addition laws and substituting the given conditions. The ratio of P to Q is sqrt(5 - 2*sqrt(5)).
Step-by-step explanation:
The problem is dealing with the magnitudes of vectors P and Q and their respective inclination. In vector physics, we make use of vector addition laws to solve such problems. We know when two vectors are inclined at an angle theta, the resultant R is given by the square root of (P² + Q² + 2PQcosθ). According to the problem, R = 2P when θ is being used.
Next, when the inclination is changed to 180-θ, the problem states the magnitude of the resultant R is halved, which means R becomes P. Let's substitute these variables into our equation to find the ratio of P to Q.
After some algebraic manipulations, we obtain the ratio P/Q = sqrt(5-2sqrt(5)). Although the formula might seem complicated, we can interpret the ratio as representing the proportional relationship between the magnitudes of the two vectors P and Q, under the conditions specified.
Learn more about Vector magnitudes