Two vectors `veca` and `vecb` add to give a resultant `vecc=veca+vecb`. In which of these cases angle between `veca` and `vecb` is maximum: (a,b,c represent the magnitudes of respective vectors )

Options

A. `c=a+b`
B. `c^(2)=a^(2)+b^(2)`
C. `c=a-b`
D. can not be determined.

Answer

To determine the case in which the angle between the two vectors \(\vec{a}\) and \(\vec{b}\) is maximum, we can analyze the given conditions using vector addition and the cosine rule. Let's break down the solution step by step.

### Step 1: Understand the vector addition
We have two vectors \(\vec{a}\) and \(\vec{b}\) that add to give a resultant vector \(\vec{c}\):
\[ \vec{c} = \vec{a} + \vec{b} \]
We want to find the angle \(\theta\) between \(\vec{a}\) and \(\vec{b}\) for different cases.

### Step 2: Square both sides
Squaring both sides of the equation gives:
\[ \vec{c} \cdot \vec{c} = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) \]
This can be expanded using the dot product:
\[ c^2 = a^2 + b^2 + 2 \vec{a} \cdot \vec{b} \]

### Step 3: Use the dot product formula
The dot product \(\vec{a} \cdot \vec{b}\) can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = ab \cos \theta \]
Substituting this into our equation gives:
\[ c^2 = a^2 + b^2 + 2ab \cos \theta \]

### Step 4: Analyze the given cases
We will analyze three cases based on the options provided:

1. **Case A**: \(c = a + b\)
- Here, \(\theta = 0^\circ\) (the vectors are in the same direction).
- Substituting \(\theta = 0\):
\[ c^2 = a^2 + b^2 + 2ab \cdot 1 = (a + b)^2 \]

2. **Case B**: \(c^2 = a^2 + b^2\)
- This occurs when \(\theta = 90^\circ\) (the vectors are perpendicular).
- Substituting \(\theta = 90^\circ\):
\[ c^2 = a^2 + b^2 + 2ab \cdot 0 = a^2 + b^2 \]

3. **Case C**: \(c = a - b\)
- This occurs when \(\theta = 180^\circ\) (the vectors are in opposite directions).
- Substituting \(\theta = 180^\circ\):
\[ c^2 = a^2 + b^2 + 2ab \cdot (-1) = a^2 + b^2 - 2ab = (a - b)^2 \]

### Step 5: Determine the maximum angle
From the analysis:
- In Case A, \(\theta = 0^\circ\)
- In Case B, \(\theta = 90^\circ\)
- In Case C, \(\theta = 180^\circ\)

The maximum angle between the vectors \(\vec{a}\) and \(\vec{b}\) occurs in Case C, where \(\theta = 180^\circ\).

### Conclusion
Thus, the angle between \(\vec{a}\) and \(\vec{b}\) is maximum in Case C, where \(c = a - b\).