For what angle between the two vectors, their resultant is maximum?

Options

A. `180^(@)`
B. zero
C. `90^(@)`
D. `45^(@)`

Explanation

To find the angle between two vectors for which their resultant is maximum, we can follow these steps:

### Step 1: Understand the Resultant of Two Vectors
When two vectors **A** and **B** are added, the magnitude of the resultant vector **R** can be expressed using the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos(\alpha)} \]
where \( \alpha \) is the angle between the two vectors.

### Step 2: Identify the Conditions for Maximum Resultant
To maximize the resultant \( R \), we need to maximize the expression under the square root, which is \( A^2 + B^2 + 2AB \cos(\alpha) \).

### Step 3: Analyze the Cosine Function
The term \( \cos(\alpha) \) varies between -1 and 1. To maximize the resultant, we need to maximize \( 2AB \cos(\alpha) \). The maximum value of \( \cos(\alpha) \) is 1, which occurs when \( \alpha = 0^\circ \).

### Step 4: Substitute the Maximum Cosine Value
When \( \alpha = 0^\circ \):
\[ R = \sqrt{A^2 + B^2 + 2AB \cdot 1} \]
This simplifies to:
\[ R = \sqrt{(A + B)^2} = A + B \]
This shows that the resultant is maximized when the two vectors are in the same direction.

### Step 5: Conclusion
Thus, the angle between the two vectors for which their resultant is maximum is:
\[ \alpha = 0^\circ \]