It is given that `vecR=vecP+vecQ` . Angle between vectors `vecP and vecQ` is `120^(@)` . Select the correct option .

Options

A. R must be grater than `|P-Q|`
B. R must be less than `|P-Q|`
C. R must be equal to `|P-Q|`
D. None of these

Explanation

To solve the problem where it is given that \(\vec{R} = \vec{P} + \vec{Q}\) and the angle between vectors \(\vec{P}\) and \(\vec{Q}\) is \(120^\circ\), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Vector Addition**:
The resultant vector \(\vec{R}\) can be found using the formula for the magnitude of the resultant of two vectors:
\[ |\vec{R}| = \sqrt{|\vec{P}|^2 + |\vec{Q}|^2 + 2 |\vec{P}| |\vec{Q}| \cos \theta} \]
where \(\theta\) is the angle between the two vectors.

2. **Substitute the Given Values**:
Here, \(\theta = 120^\circ\). The cosine of \(120^\circ\) is:
\[ \cos(120^\circ) = -\frac{1}{2} \]
Therefore, substituting this into the formula gives:
\[ |\vec{R}| = \sqrt{|\vec{P}|^2 + |\vec{Q}|^2 + 2 |\vec{P}| |\vec{Q}| \left(-\frac{1}{2}\right)} \]

3. **Simplify the Expression**:
This simplifies to:
\[ |\vec{R}| = \sqrt{|\vec{P}|^2 + |\vec{Q}|^2 - |\vec{P}| |\vec{Q}|} \]

4. **Rearranging the Terms**:
We can rearrange the expression inside the square root:
\[ |\vec{R}| = \sqrt{|\vec{P}|^2 + |\vec{Q}|^2 - |\vec{P}| |\vec{Q}|} \]

5. **Analyze the Resultant**:
To understand the relationship between \(|\vec{R}|\), \(|\vec{P}|\), and \(|\vec{Q}|\), we can compare \(|\vec{R}|\) with \(|\vec{P}| - |\vec{Q}|\):
\[ |\vec{R}|^2 = |\vec{P}|^2 + |\vec{Q}|^2 - |\vec{P}| |\vec{Q}| \]
We need to show that \(|\vec{R}| > ||\vec{P}| - |\vec{Q}||\).

6. **Using the Triangle Inequality**:
From the triangle inequality, we know that the magnitude of the resultant vector must be greater than the difference of the magnitudes of the two vectors:
\[ |\vec{R}| > ||\vec{P}| - |\vec{Q}|| \]

7. **Conclusion**:
Thus, we conclude that the magnitude of the resultant vector \(|\vec{R}|\) is greater than the absolute difference of the magnitudes of \(|\vec{P}|\) and \(|\vec{Q}|\).

### Final Answer
The correct option is that \(|\vec{R}| > ||\vec{P}| - |\vec{Q}||\).