If `|vecA+vecB|=|vecA|+|vecB|`, then angle between `vecA` and `vecB` will be

Options

A. `90^@`
B. `120^@`
C. `0^@`
D. `60^@`

Explanation

To solve the problem, we need to analyze the given condition:

**Given:**
\(|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|\)

This condition implies that the resultant vector formed by adding vectors \(\vec{A}\) and \(\vec{B}\) has the same magnitude as the sum of the magnitudes of \(\vec{A}\) and \(\vec{B}\). This situation occurs only when the two vectors are in the same direction.

### Step-by-Step Solution:

1. **Understanding the Magnitude of the Resultant Vector:**
The magnitude of the resultant vector \(\vec{R} = \vec{A} + \vec{B}\) can be expressed using the formula:
\[ |\vec{R}| = |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta} \]
where \(\theta\) is the angle between vectors \(\vec{A}\) and \(\vec{B}\).

2. **Setting Up the Equation:**
According to the problem, we have:
\[ |\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}| \]
Therefore, we can write:
\[ \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta} = |\vec{A}| + |\vec{B}| \]

3. **Squaring Both Sides:**
To eliminate the square root, we square both sides:
\[ |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta = (|\vec{A}| + |\vec{B}|)^2 \]

4. **Expanding the Right Side:**
Expanding the right-hand side gives:
\[ (|\vec{A}| + |\vec{B}|)^2 = |\vec{A}|^2 + 2 |\vec{A}| |\vec{B}| + |\vec{B}|^2 \]

5. **Setting the Equations Equal:**
Now we have:
\[ |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta = |\vec{A}|^2 + 2 |\vec{A}| |\vec{B}| + |\vec{B}|^2 \]

6. **Canceling Common Terms:**
Canceling \( |\vec{A}|^2 \) and \( |\vec{B}|^2 \) from both sides results in:
\[ 2 |\vec{A}| |\vec{B}| \cos \theta = 2 |\vec{A}| |\vec{B}| \]

7. **Dividing by \(2 |\vec{A}| |\vec{B}|\):**
Assuming \( |\vec{A}| \neq 0 \) and \( |\vec{B}| \neq 0 \), we can divide both sides by \(2 |\vec{A}| |\vec{B}|\):
\[ \cos \theta = 1 \]

8. **Finding the Angle:**
The equation \( \cos \theta = 1 \) implies:
\[ \theta = 0^\circ \]

### Conclusion:
The angle between vectors \(\vec{A}\) and \(\vec{B}\) is \(0^\circ\), meaning they are in the same direction.