If `vec(a).vec(b)=|vec(a)xx vec(b)|`, then angle between vector `vec(a)` and vector `vec(b)` is :

Options

A. `(pi)/(2)`
B. `(pi)/(6)`
C. `(pi)/(4)`
D. `(pi)/(3)`

Answer

To solve the problem, we need to find the angle between two vectors \( \vec{a} \) and \( \vec{b} \) given that \( \vec{a} \cdot \vec{b} = |\vec{a} \times \vec{b}| \).

### Step-by-step Solution:

1. **Understanding the Dot Product and Cross Product**:
The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
where \( \theta \) is the angle between the two vectors.

The magnitude of the cross product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]

2. **Setting up the Equation**:
According to the problem, we have:
\[ \vec{a} \cdot \vec{b} = |\vec{a} \times \vec{b}| \]
Substituting the formulas for the dot product and the magnitude of the cross product, we get:
\[ |\vec{a}| |\vec{b}| \cos \theta = |\vec{a}| |\vec{b}| \sin \theta \]

3. **Simplifying the Equation**:
Assuming \( |\vec{a}| \) and \( |\vec{b}| \) are not zero, we can divide both sides by \( |\vec{a}| |\vec{b}| \):
\[ \cos \theta = \sin \theta \]

4. **Using Trigonometric Identities**:
We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). From the equation \( \cos \theta = \sin \theta \), we can write:
\[ \tan \theta = 1 \]

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

### Final Answer:
The angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{4} \) radians or \( 45^\circ \).

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