Which of the following is the correct formula for the distance between the parallel lines l1 and l2?

(a) d=\left|\frac{\vec{a_2}+\vec{a_1})×(\vec{a_2}-\vec{a_1})}{|\vec{b}|}\right |

(b) d^2=\left|\frac{\vec{b}×(\vec{a_2}-\vec{a_1})}{|\vec{b}|}\right |

(c) 2d=\left|\frac{\vec{b}×(\vec{a_2}-\vec{a_1})}{|\vec{b}|}\right |

(d) d=\left|\frac{\vec{b}×(\vec{a_2}-\vec{a_1})}{|\vec{b}|}\right |

This question was addressed to me in examination.

I want to ask this question from Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12

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Correct option is (d) d=\left|\frac{\vec{b}×(\vec{a_2}-\vec{a_1})}{|\vec{b}|}\right |

The explanation is: If l1 and l2 are two parallel lines, then they are coplanar and hence can be represented by the following equations

\vec{r}=\vec{a_1}+λ\vec{b}

\vec{r}=\vec{a_2}+μ\vec{b}

Then the distance between the lines is given by

d=\left|\frac{\vec{b}×(\vec{a_2}-\vec{a_1})}{|\vec{b}|}\right |