What is the equation of the plane containing the parallel lines
$r = a + t c, r = b + p c$ .

r \cdot (b - a) \times c = [a b c]

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r \cdot (a - b) \times c = [a b c]

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r \cdot (b - a) \times c = - [a b c]

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r c \times (b - a) \times c = [a b c]

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Solution

The correct option is A $r \cdot (b - a) \times c = [a b c]$
The direction vector of both the lines is $\to c$
Hence the normal to the required plane will be a vector $\to n$ which perpendicular to $\to A B$ and $\to c$ .
Therefore the normal vector will be
$r . (\to A B \times \to c) = [a b c]$
Or
$r . ((\to b - \to a) . \times \to c) = [a b c]$
Or
$r . ((\to a - \to b) \times \to c) = - [a b c]$

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What is the equation of the plane containing the parallel linesr=a+tc, r=b+pc.

What is the equation of the plane containing the parallel lines
$r = a + t c, r = b + p c$ .