Question

What is the distance between \[\vec r=(\hat i+2\hat j-4\hat k)+\lambda(2\hat i+3\hat j+6\hat k)\] and \[\vec r=(3\hat i+3\hat j-5\hat k)+\mu(2\hat i+3\hat j+6\hat k)\]?

Options

• \[\frac{293}{7}\]

• \[\frac{7}{\sqrt{293}}\]

• \[\frac{\sqrt{293}}{7}\]

• \[\frac{\sqrt{49}}{\sqrt{293}}\]

Answer 1

The lines have the common direction vector \[\vec b=2\hat i+3\hat j+6\hat k\], so they are parallel. Since \[|\vec b|=\sqrt{49}=7\] and the cross-product magnitude is \[\sqrt{293}\], the distance is \[\frac{\sqrt{293}}{7}\].