`vecP, vecQ, vecR, vecS` are vector of equal magnitude. If `vecP + vecQ - vecR=0` angle between `vecP` and `vecQ` is `theta_(1)` . If `vecP + vecQ - vecS =0` angle between `vecP` and `vecS` is `theta_(2)` . The ratio of `theta_(1)` to `theta_(2)` is

Options

A. `1:2`
B. `2:1`
C. `1:1`
D. `1:sqrt(3)`

Answer

To solve the problem, we need to analyze the given vectors and the relationships between them step by step.

### Step 1: Understanding the Given Vectors
We are given four vectors: \(\vec{P}\), \(\vec{Q}\), \(\vec{R}\), and \(\vec{S}\), all of equal magnitude. The relationships provided are:
1. \(\vec{P} + \vec{Q} - \vec{R} = 0\)
2. \(\vec{P} + \vec{Q} - \vec{S} = 0\)

From these equations, we can rearrange them to express \(\vec{R}\) and \(\vec{S}\):
- \(\vec{R} = \vec{P} + \vec{Q}\)
- \(\vec{S} = \vec{P} + \vec{Q}\)

### Step 2: Analyzing the First Equation
From the first equation, we can visualize the vectors \(\vec{P}\) and \(\vec{Q}\) as two sides of a triangle with \(\vec{R}\) as the third side. Since all vectors are of equal magnitude, we can represent them as sides of an equilateral triangle.

### Step 3: Finding the Angle \(\theta_1\)
In an equilateral triangle, each angle is \(60^\circ\). However, since we are looking for the angle between \(\vec{P}\) and \(\vec{Q}\) when considering the vector addition, we note that:
- The angle between \(\vec{P}\) and \(\vec{Q}\) is \(120^\circ\) (since the vectors point in opposite directions forming a triangle).

Thus, we have:
\[ \theta_1 = 120^\circ \]

### Step 4: Analyzing the Second Equation
For the second equation, \(\vec{P} + \vec{Q} = \vec{S}\), we can again visualize this as a triangle. Since \(\vec{S}\) is also equal in magnitude to \(\vec{R}\), we can conclude that \(\vec{S}\) forms another triangle with \(\vec{P}\) and \(\vec{Q}\).

### Step 5: Finding the Angle \(\theta_2\)
The angle between \(\vec{P}\) and \(\vec{S}\) is formed similarly to the previous case. Since \(\vec{S}\) is the resultant of \(\vec{P}\) and \(\vec{Q}\), the angle between \(\vec{P}\) and \(\vec{S}\) is:
\[ \theta_2 = 60^\circ \]

### Step 6: Finding the Ratio of Angles
Now that we have both angles:
- \(\theta_1 = 120^\circ\)
- \(\theta_2 = 60^\circ\)

We can find the ratio \(\frac{\theta_1}{\theta_2}\):
\[ \frac{\theta_1}{\theta_2} = \frac{120^\circ}{60^\circ} = 2 \]

Thus, the ratio of \(\theta_1\) to \(\theta_2\) is:
\[ \text{Ratio} = 2:1 \]

### Final Answer
The ratio of \(\theta_1\) to \(\theta_2\) is \(2:1\).

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