a,b are the magnitudes of vectors `veca` & `vecb` . If `vecaxxvecb=0` the value of `veca.vecb` is

Options

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

Answer

To solve the problem, we need to analyze the given information about the vectors \(\vec{a}\) and \(\vec{b}\).

### Step-by-Step Solution:

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

2. **Interpret the Cross Product**:
The cross product of two vectors is zero if and only if the vectors are parallel or one of them is the zero vector. Mathematically, this can be expressed as:
\[ |\vec{a}| |\vec{b}| \sin \theta = 0 \]
where \(\theta\) is the angle between the two vectors.

3. **Analyze the Condition**:
Since the magnitudes of the vectors \(\vec{a}\) and \(\vec{b}\) are given as \(a\) and \(b\) respectively, we can conclude that:
\[ a \cdot b \cdot \sin \theta = 0 \]
This implies that either \(a = 0\), \(b = 0\), or \(\sin \theta = 0\). Since we are looking for the dot product, we will focus on the case where \(\sin \theta = 0\).

4. **Determine the Angle**:
If \(\sin \theta = 0\), then \(\theta\) must be either \(0^\circ\) or \(180^\circ\). This means that the vectors \(\vec{a}\) and \(\vec{b}\) are in the same direction or opposite directions.

5. **Calculate the Dot Product**:
The dot product of two vectors is given by the formula:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
Since we have determined that \(\theta = 0^\circ\) or \(\theta = 180^\circ\):
- If \(\theta = 0^\circ\), then \(\cos 0^\circ = 1\).
- If \(\theta = 180^\circ\), then \(\cos 180^\circ = -1\).

Therefore, we can express the dot product as:
\[ \vec{a} \cdot \vec{b} = ab \cdot \cos \theta \]
Depending on the angle:
- For \(\theta = 0^\circ\): \(\vec{a} \cdot \vec{b} = ab \cdot 1 = ab\)
- For \(\theta = 180^\circ\): \(\vec{a} \cdot \vec{b} = ab \cdot (-1) = -ab\)

6. **Final Answer**:
Since the problem does not specify the direction of the vectors, the value of \(\vec{a} \cdot \vec{b}\) can be either \(ab\) or \(-ab\). However, typically, we take the positive value when discussing magnitudes, so we conclude:
\[ \vec{a} \cdot \vec{b} = ab \]