Question

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

Answer 1

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