Two vectors `vec(A)and vec(B)` are at right angles to each other then

Options

A. `vec(A) + vec(B) = 0`
B. `vec(A) - vec(B) = 0`
C. `vec(A) xx vec(B) = 0`
D. `vec(A) .vec(B) = 0`

Answer

To solve the problem where two vectors \(\vec{A}\) and \(\vec{B}\) are such that the magnitude of their dot product is equal to the magnitude of their cross product, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Given Condition**:
We are given that:
\[ |\vec{A} \cdot \vec{B}| = |\vec{A} \times \vec{B}| \]

2. **Use the Formulas for Dot and Cross Products**:
The dot product of two vectors can be expressed as:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]
The cross product of two vectors can be expressed as:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]

3. **Set the Two Expressions Equal**:
From the given condition, we can write:
\[ |\vec{A}| |\vec{B}| \cos \theta = |\vec{A}| |\vec{B}| \sin \theta \]

4. **Cancel Common Terms**:
Assuming \( |\vec{A}| \) and \( |\vec{B}| \) are not zero, we can divide both sides by \( |\vec{A}| |\vec{B}| \):
\[ \cos \theta = \sin \theta \]

5. **Use the Identity**:
The equation \( \cos \theta = \sin \theta \) implies:
\[ \tan \theta = 1 \]

6. **Find the Angle**:
The angle \( \theta \) for which \( \tan \theta = 1 \) is:
\[ \theta = 45^\circ \quad \text{or} \quad \theta = \frac{\pi}{4} \text{ radians} \]

### Final Answer
The angle between the vectors \(\vec{A}\) and \(\vec{B}\) is \(45^\circ\).

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