The two lines `vecr=veca+veclamda(vecbxxvecc) and vecr=vecb+mu(veccxxveca)` intersect at a point where `veclamda and mu` are scalars then

Options

A. `vecaxxvecc=vecbxxvecc`
B. `veca.vecc=vecb.vecc`
C. `vecbxxveca=veccxxveca`
D. none of these

Explanation

To solve the problem, we need to analyze the given lines and their intersection conditions. Let's break it down step by step.

### Step 1: Understand the given lines
We have two lines represented as:
1. \(\vec{r} = \vec{a} + \lambda (\vec{b} \times \vec{c})\)
2. \(\vec{r} = \vec{b} + \mu (\vec{c} \times \vec{a})\)

Here, \(\lambda\) and \(\mu\) are scalar parameters.

### Step 2: Condition for intersection
For the two lines to intersect, the vectors formed by the points on the lines must be coplanar. This means that the vector from point \(\vec{b}\) to point \(\vec{a}\) and the direction vectors of the lines must be coplanar.

### Step 3: Define the vectors
Let:
- \(\vec{d_1} = \vec{b} - \vec{a}\) (the vector from \(\vec{a}\) to \(\vec{b}\))
- \(\vec{d_2} = \vec{b} \times \vec{c}\) (the direction vector of the first line)
- \(\vec{d_3} = \vec{c} \times \vec{a}\) (the direction vector of the second line)

### Step 4: Apply the coplanarity condition
The vectors \(\vec{d_1}\), \(\vec{d_2}\), and \(\vec{d_3}\) must be coplanar. This can be expressed using the scalar triple product:
\[ (\vec{a} - \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} - \vec{a}) \cdot (\vec{c} \times \vec{a}) = 0 \]

### Step 5: Expand the scalar triple product
Using the properties of the scalar triple product, we can rewrite the condition:
\[ (\vec{a} - \vec{b}) \cdot (\vec{b} \times \vec{c}) = 0 \]
This implies that:
\[ \vec{a} - \vec{b} \text{ is perpendicular to } \vec{b} \times \vec{c} \]

### Step 6: Analyze the options
Now we analyze the options given in the problem:
1. \(\vec{a} \times \vec{c} = \vec{b} \times \vec{c}\)
2. \(\vec{a} \cdot \vec{c} = \vec{b} \cdot \vec{c}\)
3. \(\vec{b} \times \vec{a} = \vec{c} \times \vec{a}\)
4. None of these

From our analysis, we found that the condition simplifies to:
\[ \vec{a} \cdot \vec{c} = \vec{b} \cdot \vec{c} \]
Thus, the correct answer is option 2.

### Final Answer
The correct option is:
**Option B: \(\vec{a} \cdot \vec{c} = \vec{b} \cdot \vec{c}\)**