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

Category: QuestionsWhich of the following is the correct formula for the distance between the parallel lines l1 and l2?
Editor">Editor Staff asked 11 months ago

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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1 Answers
Editor">Editor Staff answered 11 months ago

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 |