When two vectors of magnitudes P and Q are inclined at an angle `theta`, the magnitudes of their resultant is 2P. When the inclination is changed to `180^(@)-theta`, the magnitudes of the resultant is halved. Find the ratio of `P` and `Q`.

Explanation

To solve the problem, we will use the properties of vectors and the law of cosines. Let's break it down step by step.

### Step 1: Write the equation for the resultant of two vectors at angle θ
When two vectors of magnitudes P and Q are inclined at an angle θ, the magnitude of their resultant R can be given by the formula:

\[ R^2 = P^2 + Q^2 + 2PQ \cos \theta \]

According to the problem, the magnitude of the resultant is 2P. Therefore, we can write:

\[ (2P)^2 = P^2 + Q^2 + 2PQ \cos \theta \]

This simplifies to:

\[ 4P^2 = P^2 + Q^2 + 2PQ \cos \theta \]

### Step 2: Rearranging the equation
Now, rearranging the above equation gives us:

\[ 4P^2 - P^2 - Q^2 = 2PQ \cos \theta \]

This simplifies to:

\[ 3P^2 - Q^2 = 2PQ \cos \theta \quad \text{(Equation 1)} \]

### Step 3: Write the equation for the resultant of two vectors at angle 180° - θ
When the angle is changed to \(180^\circ - \theta\), the cosine term changes to \(-\cos \theta\). Thus, the resultant R' can be expressed as:

\[ R'^2 = P^2 + Q^2 - 2PQ \cos \theta \]

According to the problem, the magnitude of the resultant is halved, which means:

\[ \left(\frac{2P}{2}\right)^2 = P^2 + Q^2 - 2PQ \cos \theta \]

This simplifies to:

\[ P^2 = P^2 + Q^2 - 2PQ \cos \theta \]

### Step 4: Rearranging the second equation
Rearranging gives us:

\[ P^2 - P^2 - Q^2 = -2PQ \cos \theta \]

This simplifies to:

\[ -Q^2 = -2PQ \cos \theta \]

or

\[ Q^2 = 2PQ \cos \theta \quad \text{(Equation 2)} \]

### Step 5: Solve the two equations
Now we have two equations:

1. \(3P^2 - Q^2 = 2PQ \cos \theta\) (Equation 1)
2. \(Q^2 = 2PQ \cos \theta\) (Equation 2)

Substituting Equation 2 into Equation 1:

\[ 3P^2 - 2PQ \cos \theta = 2PQ \cos \theta \]

This simplifies to:

\[ 3P^2 = 4PQ \cos \theta \]

### Step 6: Express Q in terms of P
Rearranging gives:

\[ Q = \frac{3P^2}{4P \cos \theta} = \frac{3P}{4 \cos \theta} \]

### Step 7: Find the ratio of P and Q
Now, we can find the ratio of P to Q:

\[ \frac{P}{Q} = \frac{P}{\frac{3P}{4 \cos \theta}} = \frac{4 \cos \theta}{3} \]

### Conclusion
Thus, the ratio of P to Q is:

\[ \frac{P}{Q} = \frac{4 \cos \theta}{3} \]