a, b are the magnitudes of vectors `vec(a)` & `vec(b)`. If `vec(a)xx vec(b)=0` the value of `vec(a).vec(b)` is

Options

A. 0
B. `sqrt(ab)`
C. ab
D. a/b

Answer

To solve the problem, we need to analyze the relationship between the cross product and dot product of two vectors, given that the cross product is equal to zero.

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given that the cross product of two vectors \(\vec{a}\) and \(\vec{b}\) is zero:
\[ \vec{a} \times \vec{b} = 0 \]

2. **Interpret the Cross Product**:
The cross product of two vectors is defined as:
\[ \vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \hat{n} \]
where \(\theta\) is the angle between the two vectors, and \(\hat{n}\) is the unit vector perpendicular to the plane containing \(\vec{a}\) and \(\vec{b}\).

3. **Set the Cross Product to Zero**:
Since \(\vec{a} \times \vec{b} = 0\), it implies:
\[ |\vec{a}| |\vec{b}| \sin \theta = 0 \]
For this product to be zero, either \(|\vec{a}| = 0\), \(|\vec{b}| = 0\), or \(\sin \theta = 0\). Since we are considering non-zero vectors, we conclude:
\[ \sin \theta = 0 \]

4. **Determine the Angle \(\theta\)**:
The condition \(\sin \theta = 0\) implies that:
\[ \theta = 0^\circ \text{ or } \theta = 180^\circ \]
This means the vectors \(\vec{a}\) and \(\vec{b}\) are either parallel or anti-parallel.

5. **Calculate the Dot Product**:
The dot product of two vectors is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
Substituting \(\theta = 0^\circ\) (for parallel vectors):
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(0^\circ) = |\vec{a}| |\vec{b}| \cdot 1 = |\vec{a}| |\vec{b}| \]
If \(\theta = 180^\circ\) (for anti-parallel vectors):
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(180^\circ) = |\vec{a}| |\vec{b}| \cdot (-1) = -|\vec{a}| |\vec{b}| \]

6. **Final Result**:
Since the magnitudes of \(\vec{a}\) and \(\vec{b}\) are given as \(a\) and \(b\) respectively, we conclude that:
\[ \vec{a} \cdot \vec{b} = ab \text{ or } -ab \]
However, the problem does not specify the direction of the vectors, so we can state:
\[ \vec{a} \cdot \vec{b} = ab \]

### Conclusion:
The value of \(\vec{a} \cdot \vec{b}\) is \(ab\).