The resultant of `vec(P)` and `vec(Q)` is `vec(R)`. If `vec(Q)` is doubled, `vec(R)` is doubled, when `vec(Q)` is reversed, `vec(R)` is again doubled. Find P:Q:R.

Explanation

To solve the problem, we need to analyze the relationships between the vectors \( \vec{P} \), \( \vec{Q} \), and \( \vec{R} \) based on the given conditions. Let's break it down step by step.

### Step 1: Establish the initial relationship
We know that the resultant of vectors \( \vec{P} \) and \( \vec{Q} \) is \( \vec{R} \). Mathematically, this is expressed as:
\[ R^2 = P^2 + Q^2 + 2PQ \cos \theta \]
where \( P \), \( Q \), and \( R \) are the magnitudes of vectors \( \vec{P} \), \( \vec{Q} \), and \( \vec{R} \) respectively, and \( \theta \) is the angle between \( \vec{P} \) and \( \vec{Q} \).
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