The vector equation of the plane passing through the origin and the line of intersection of the planes `vecr.veca=lamdaandvecr.vecb=mu` is

Options

A. `vecr.(lamdaveca-muvecb)=0`
B. `vecr.(lamdavecb-muveca)=0`
C. `vecr.(lamdaveca+muvecb)=0`
D. `vecr.(lamdavecb+muveca)=0`

Explanation

To find the vector equation of the plane passing through the origin and the line of intersection of the planes defined by the equations \( \vec{r} \cdot \vec{a} = \lambda \) and \( \vec{r} \cdot \vec{b} = \mu \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Planes**:
The equations \( \vec{r} \cdot \vec{a} = \lambda \) and \( \vec{r} \cdot \vec{b} = \mu \) represent two planes in three-dimensional space. The line of intersection of these two planes can be described by a linear combination of these two equations.

2. **Equation of the Plane through the Intersection**:
The general equation of a plane passing through the line of intersection of two planes can be expressed as:
\[ \vec{r} \cdot \vec{a} - \lambda + k(\vec{r} \cdot \vec{b} - \mu) = 0 \]
where \( k \) is a constant.

3. **Condition for the Plane to Pass Through the Origin**:
Since the plane must pass through the origin, we substitute \( \vec{r} = \vec{0} \) into the equation:
\[ \vec{0} \cdot \vec{a} - \lambda + k(\vec{0} \cdot \vec{b} - \mu) = 0 \]
This simplifies to:
\[ -\lambda - k\mu = 0 \]
From this, we can solve for \( k \):
\[ k = -\frac{\lambda}{\mu} \]

4. **Substituting \( k \) Back into the Plane Equation**:
Now substitute \( k \) back into the equation of the plane:
\[ \vec{r} \cdot \vec{a} - \lambda - \frac{-\lambda}{\mu}(\vec{r} \cdot \vec{b} - \mu) = 0 \]
This simplifies to:
\[ \vec{r} \cdot \vec{a} - \lambda + \frac{\lambda}{\mu} \vec{r} \cdot \vec{b} - \lambda = 0 \]
Rearranging gives:
\[ \vec{r} \cdot \vec{a} + \frac{\lambda}{\mu} \vec{r} \cdot \vec{b} - 2\lambda = 0 \]

5. **Final Form of the Plane Equation**:
We can factor out \( \vec{r} \):
\[ \vec{r} \cdot \left( \vec{a} + \frac{\lambda}{\mu} \vec{b} \right) - 2\lambda = 0 \]
Thus, the final vector equation of the plane is:
\[ \vec{r} \cdot \left( \mu \vec{a} - \lambda \vec{b} \right) = 0 \]

### Final Answer:
The vector equation of the plane is:
\[ \vec{r} \cdot (\mu \vec{a} - \lambda \vec{b}) = 0 \]