If `vec(c )=vec(a)-vec(b)`, which of the following relations among magnitudes cannot be achieved by any choice of `vec(a)` and `vec(b)` ?

Options

A. `c=a-b`
B. `c gt a, c gt b`
C. `c=a+b`
D. `c lt a-b`

Explanation

To solve the problem, we start with the vector equation given:

\[ \vec{c} = \vec{a} - \vec{b} \]

### Step 1: Understanding the Magnitude of the Resultant Vector
The magnitude of vector \(\vec{c}\) can be expressed using the formula for the magnitude of the difference of two vectors:

\[ |\vec{c}| = |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2 |\vec{a}| |\vec{b}| \cos \theta} \]

where \(\theta\) is the angle between vectors \(\vec{a}\) and \(\vec{b}\).

### Step 2: Establishing Maximum and Minimum Values
From the formula, we can analyze the maximum and minimum possible values of \(|\vec{c}|\):

1. **Maximum Value**: This occurs when \(\cos \theta = -1\) (i.e., when \(\vec{a}\) and \(\vec{b}\) are in opposite directions):
\[ |\vec{c}|_{\text{max}} = |\vec{a}| + |\vec{b}| \]

2. **Minimum Value**: This occurs when \(\cos \theta = 1\) (i.e., when \(\vec{a}\) and \(\vec{b}\) are in the same direction):
\[ |\vec{c}|_{\text{min}} = ||\vec{a}| - |\vec{b}|| \]

### Step 3: Analyzing the Options
Now, we need to determine which of the given relations among the magnitudes cannot be achieved by any choice of \(\vec{a}\) and \(\vec{b}\).

- The possible range for \(|\vec{c}|\) is:
\[ ||\vec{a}| - |\vec{b}|| \leq |\vec{c}| \leq |\vec{a}| + |\vec{b}| \]

### Step 4: Conclusion
Based on the above analysis, any relation that falls outside this range cannot be achieved. Therefore, the relation that cannot be achieved is:

\[ |\vec{c}| < ||\vec{a}| - |\vec{b}|| \]

This means that the option stating that \(|\vec{c}|\) can be less than the absolute difference of the magnitudes of \(\vec{a}\) and \(\vec{b}\) is the one that cannot be achieved.

### Final Answer
The relation among magnitudes that cannot be achieved by any choice of \(\vec{a}\) and \(\vec{b}\) is:

\[ |\vec{c}| < ||\vec{a}| - |\vec{b}|| \]

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