For the resultant of two vectors to be maximum , what must be the angle between them ?

Explanation

To determine the angle between two vectors for their resultant to be maximum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Resultant of Two Vectors**:
The resultant \( R \) of two vectors \( \vec{A} \) and \( \vec{B} \) can be expressed using the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
where \( A \) and \( B \) are the magnitudes of vectors \( \vec{A} \) and \( \vec{B} \), respectively, and \( \theta \) is the angle between them.

2. **Identifying the Variables**:
In this equation, \( A \) and \( B \) are constants (the magnitudes of the vectors), while \( \cos \theta \) is a variable that can take values between -1 and 1.

3. **Maximizing the Resultant**:
To maximize the resultant \( R \), we need to maximize the term \( 2AB \cos \theta \). The maximum value of \( \cos \theta \) occurs when \( \cos \theta = 1 \).

4. **Finding the Angle**:
The condition \( \cos \theta = 1 \) occurs when \( \theta = 0^\circ \). This means that the two vectors \( \vec{A} \) and \( \vec{B} \) must be in the same direction (parallel) for the resultant to be maximum.

5. **Conclusion**:
Therefore, for the resultant of two vectors to be maximum, the angle \( \theta \) between them must be:
\[ \theta = 0^\circ \]

### Final Answer:
The angle between the two vectors must be \( 0^\circ \) for their resultant to be maximum.

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