Question

Which of the below given is the correct formula for the distance between two skew lines l1 and l2?

(a) d=\(\left |\frac{(\vec{b_1}×\vec{b_2}).(a_2-a_1)}{|\vec{b_1}×\vec{b_2}|}\right |\)

(b) 2d=\(\left |\frac{(\vec{b_1}-\vec{b_2}).(a_2-a_1)}{|\vec{b_1}-\vec{b_2}|}\right |\)

(c) d=\(\left |\frac{(\vec{b_1}×\vec{b_2}).(a_2.a_1)}{3|\vec{b_1}×\vec{b_2}|}\right |\)

(d) d^2=\(\left |\frac{(\vec{b_1}×\vec{b_2}).(a_2-a_1)}{|\vec{b_1}-\vec{b_2}|}\right |\)

This question was addressed to me during an online interview.

I need to ask this question from Three Dimensional Geometry topic in division Three Dimensional Geometry of Mathematics – Class 12

Answer 1

The correct answer is (a) d=\(\left |\frac{(\vec{b_1}×\vec{b_2}).(a_2-a_1)}{|\vec{b_1}×\vec{b_2}|}\right |\)

Easy explanation: The distance between two lines l1 and l2 with the equations

\(\vec{r}=\vec{a_1}+λ\vec{b_1}\)

\(\vec{r}=\vec{a_2}+μ\vec{b_2}\)

Then, the distance between the two lines is given by the formula

d=\(\left |\frac{(\vec{b_1}×\vec{b_2}).(a_2-a_1)}{|\vec{b_1}×\vec{b_2}|}\right |\)