Let L₁ and L₂ be two parallel lines with the equations \(\rm \vec{r}=\vec{a}_1 +\lambda \vec{b}\) and \(\rm r=\vec{a}_2 + \mu\vec{b}\) respectively. The shortest distance between them is:

Concept:

• If two lines are parallel, then the distance between them is fixed.
• The distance between two parallel lines \(\rm \vec{r}=\vec{a}_1 +\lambda \vec{b}\) and \(\rm r=\vec{a}_2 + \mu\vec{b}\) is given by the formula: \(\rm d=\left|\dfrac{\vec{b}\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\).

Calculation:

Using the formula for the distance between two parallel lines, we can say that the distance is \(\rm d=\left|\dfrac{\vec{b}\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\).