The angle between the two vectors `veca + vecb` and `veca-vecb` is

Options

A. Between `0^(@)` and `180^(@)`
B. `90^(@)` only
C. `0^(@)` only
D. `180^(@)` only

Answer

To find the angle between the two vectors \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\), we can use the concept of the dot product of vectors. The angle \(\theta\) between two vectors \(\vec{u}\) and \(\vec{v}\) can be calculated using the formula:

\[ \cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \]

### Step 1: Define the vectors
Let:
\[ \vec{C} = \vec{a} + \vec{b} \]
\[ \vec{D} = \vec{a} - \vec{b} \]

### Step 2: Calculate the dot product \(\vec{C} \cdot \vec{D}\)
The dot product of \(\vec{C}\) and \(\vec{D}\) is given by:
\[ \vec{C} \cdot \vec{D} = (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) \]
Using the distributive property of the dot product:
\[ \vec{C} \cdot \vec{D} = \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} \]
Since \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\), we can simplify this to:
\[ \vec{C} \cdot \vec{D} = |\vec{a}|^2 - |\vec{b}|^2 \]

### Step 3: Calculate the magnitudes of \(\vec{C}\) and \(\vec{D}\)
The magnitude of \(\vec{C}\) is:
\[ |\vec{C}| = |\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b}} \]
The magnitude of \(\vec{D}\) is:
\[ |\vec{D}| = |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b}} \]

### Step 4: Substitute into the cosine formula
Now we can substitute the values into the cosine formula:
\[ \cos(\theta) = \frac{|\vec{a}|^2 - |\vec{b}|^2}{|\vec{C}| |\vec{D}|} \]

### Step 5: Analyze the angle
The angle \(\theta\) can take values between \(0^\circ\) and \(180^\circ\) depending on the values of \(|\vec{a}|\) and \(|\vec{b}|\). Thus, the angle between the vectors \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\) is indeed between \(0^\circ\) and \(180^\circ\).

### Conclusion
Therefore, the angle between the two vectors \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\) is between \(0^\circ\) and \(180^\circ\).