The maximum value of magnitude of `(vecA-vecB)` is

Options

A. A-B
B. A+B
C. `A^(2) + B^(2)`
D. `A^(2)-B^(2)`

Answer

To find the maximum value of the magnitude of the vector difference \( \vec{A} - \vec{B} \), we can follow these steps:

### Step 1: Understand the Vector Difference
The vector difference \( \vec{A} - \vec{B} \) can be interpreted geometrically. The magnitude of this difference can be maximized depending on the relative directions of the vectors \( \vec{A} \) and \( \vec{B} \).

### Step 2: Use the Triangle Inequality
According to the triangle inequality, the magnitude of the difference of two vectors can be expressed as:
\[ |\vec{A} - \vec{B}| = |\vec{A}| + |\vec{B}| \]
This equality holds when the vectors are in opposite directions.

### Step 3: Set Up the Vectors
Assume:
- \( \vec{A} = a \hat{i} \)
- \( \vec{B} = b \hat{i} \)
If \( \vec{A} \) and \( \vec{B} \) are in opposite directions, we can represent \( \vec{B} \) as:
\[ \vec{B} = -b \hat{i} \]

### Step 4: Calculate the Difference
Now, substituting these into the expression for the vector difference:
\[ \vec{A} - \vec{B} = a \hat{i} - (-b \hat{i}) = a \hat{i} + b \hat{i} = (a + b) \hat{i} \]

### Step 5: Find the Magnitude
The magnitude of the vector difference is:
\[ |\vec{A} - \vec{B}| = |(a + b) \hat{i}| = a + b \]

### Step 6: Conclusion
Thus, the maximum value of the magnitude of \( \vec{A} - \vec{B} \) occurs when the vectors are in opposite directions and is given by:
\[ \text{Maximum value} = |\vec{A}| + |\vec{B}| \]

### Final Answer
The maximum value of the magnitude of \( \vec{A} - \vec{B} \) is \( |\vec{A}| + |\vec{B}| \).

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