Lines `vecr = veca_(1) + lambda vecb and vecr = veca_(2) + svecb_` will lie in a Plane if

Options

A. `veca_(1) xxveca_(2)=0`
B. `(veca_(2)-veca_(1))xxvecb_(1) xx vecb_(2) = 0`
C. `(veca_(2)-veca_(1))cdot (vecb_(1) xx vecb_(2)) = 0`
D. `vecb_(1) xxvecb_(2)=0`

Answer

To determine whether the lines given by the equations \(\vec{r} = \vec{a_1} + \lambda \vec{b_1}\) and \(\vec{r} = \vec{a_2} + s \vec{b_2}\) lie in a plane, we can use the following conditions:

### Step 1: Understand the Conditions for Lines to Lie in a Plane
Two lines will lie in the same plane if either:
1. They intersect (the shortest distance between them is zero).
2. They are parallel (the direction ratios are proportional).

### Step 2: Condition for Intersection
For the lines to intersect, the shortest distance \(d\) between them must be zero. The formula for the shortest distance between two skew lines is given by:
\[ d = \frac{(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})}{|\vec{b_1} \times \vec{b_2}|} \]
For the lines to intersect, we require:
\[ (\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2}) = 0 \]
This means that the vector \((\vec{a_2} - \vec{a_1})\) is perpendicular to the vector \((\vec{b_1} \times \vec{b_2})\).

### Step 3: Condition for Parallel Lines
For the lines to be parallel, the direction vectors \(\vec{b_1}\) and \(\vec{b_2}\) must be proportional. This can be expressed as:
\[ \vec{b_1} \times \vec{b_2} = \vec{0} \]
This means that the direction vectors are either in the same direction or in opposite directions.

### Conclusion
Thus, the lines will lie in a plane if either of the following conditions is satisfied:
1. \((\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2}) = 0\) (they intersect).
2. \(\vec{b_1} \times \vec{b_2} = \vec{0}\) (they are parallel).

### Final Answer
The lines will lie in a plane if either of the conditions mentioned above holds true.