Given that `A=B`. What is the angle between `(vecA+vecB) and (vecA-vecB)` ?

Options

A. `30^(@) `
B. `60^(@)`
C. `90^(@)`
D. `180^(@)`

Explanation

To find the angle between the vectors \((\vec{A} + \vec{B})\) and \((\vec{A} - \vec{B})\) given that \(\vec{A} = \vec{B}\), we can follow these steps:

### Step 1: Understand the Given Information
We know that \(\vec{A} = \vec{B}\). This means that both vectors are equal in magnitude and direction.

### Step 2: Substitute \(\vec{B}\) with \(\vec{A}\)
Since \(\vec{A} = \vec{B}\), we can substitute \(\vec{B}\) in the expressions:
\[ \vec{A} + \vec{B} = \vec{A} + \vec{A} = 2\vec{A} \]
\[ \vec{A} - \vec{B} = \vec{A} - \vec{A} = \vec{0} \]

### Step 3: Analyze the Resulting Vectors
Now we have:
- \((\vec{A} + \vec{B}) = 2\vec{A}\)
- \((\vec{A} - \vec{B}) = \vec{0}\)

### Step 4: Find the Angle Between the Two Vectors
The angle \(\theta\) between two vectors can be found using the dot product formula:
\[ \vec{U} \cdot \vec{V} = |\vec{U}| |\vec{V}| \cos \theta \]
In our case, we have:
- \(\vec{U} = 2\vec{A}\)
- \(\vec{V} = \vec{0}\)

The dot product of any vector with the zero vector is zero:
\[ (2\vec{A}) \cdot \vec{0} = 0 \]

### Step 5: Set Up the Equation
Using the dot product formula:
\[ 0 = |2\vec{A}| |\vec{0}| \cos \theta \]
Since \(|\vec{0}| = 0\), the right-hand side becomes zero regardless of the value of \(\theta\).

### Step 6: Conclusion about the Angle
Since the dot product is zero, it implies that the angle between the two vectors is \(90^\circ\) (as the cosine of \(90^\circ\) is zero).

### Final Answer
Thus, the angle between \((\vec{A} + \vec{B})\) and \((\vec{A} - \vec{B})\) is:
\[ \theta = 90^\circ \]

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