The equation of the plane passing through three non - collinear points with positions vectors a,b,c, is

Options

A. `vecr.(vecaxxvecb+vecbxxvecc+veccxxveca)=0`
B. `vecrxx(vecaxxvecb+vecbxxvecc)=[(veca,vecb,vecc)]`
C. `vecr.(vecaxxvecb+vecbxxvecc+veccxxveca)+[(veca,vecb,vecc)]=0`
D. none of these

Explanation

To find the equation of the plane passing through three non-collinear points with position vectors **a**, **b**, and **c**, we can follow these steps:

### Step 1: Define the Position Vector of a Point on the Plane
Let **R** be the position vector of any point on the plane. The vectors **a**, **b**, and **c** represent three points in space.

### Step 2: Formulate the Vector Equation
The vector equation of the plane can be expressed using the position vectors of the points. We can express the vector **R** in terms of the vectors **a**, **b**, and **c**:
\[ \text{R} - \text{a} = \lambda (\text{b} - \text{a}) + \mu (\text{c} - \text{a}) \]
where \(\lambda\) and \(\mu\) are scalars.

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ \text{R} = \text{a} + \lambda (\text{b} - \text{a}) + \mu (\text{c} - \text{a}) \]

### Step 4: Use the Cross Product to Find the Normal Vector
The normal vector to the plane can be found using the cross product of two vectors lying in the plane:
\[ \text{n} = (\text{b} - \text{a}) \times (\text{c} - \text{a}) \]

### Step 5: Write the Equation of the Plane
The general equation of a plane can be written as:
\[ \text{n} \cdot (\text{R} - \text{a}) = 0 \]
Substituting the normal vector:
\[ ((\text{b} - \text{a}) \times (\text{c} - \text{a})) \cdot (\text{R} - \text{a}) = 0 \]

### Step 6: Expand the Equation
Expanding this gives:
\[ (\text{b} - \text{a}) \times (\text{c} - \text{a}) \cdot \text{R} - (\text{b} - \text{a}) \times (\text{c} - \text{a}) \cdot \text{a} = 0 \]

### Step 7: Final Form
The final equation of the plane can be expressed as:
\[ \text{R} \cdot ((\text{b} - \text{a}) \times (\text{c} - \text{a})) = \text{a} \cdot ((\text{b} - \text{a}) \times (\text{c} - \text{a})) \]

### Summary
Thus, the equation of the plane passing through the points with position vectors **a**, **b**, and **c** is:
\[ \text{R} \cdot ((\text{b} - \text{a}) \times (\text{c} - \text{a})) = \text{a} \cdot ((\text{b} - \text{a}) \times (\text{c} - \text{a})) \]