The problem involves finding the radius of the smallest sphere tangent to two skew lines defined by the parametric equations L1: x=t+1, y=2t+4, z=−3t+5 and L2: x=4t−12, y=t+5, z=t+17. The key insight is that the distance between these two lines determines the diameter of the sphere, as the unique perpendicular between skew lines is equal to this distance. Therefore, the radius of the sphere is half of this distance. Understanding the relationship between skew lines and their perpendicular distance is crucial for solving this problem.
PREREQUISITESStudents and educators in Calculus 3, particularly those focusing on three-dimensional geometry and applications involving spheres and lines. This discussion is beneficial for anyone looking to deepen their understanding of geometric relationships in higher dimensions.
hitachiin69 said:Find the radius of smallest sphere that is tangent to both the lines
L1 :x=t+1,y=2t+4,z=−3t+5 and L2 :x=4t−12,y=t+5,z=t+17.
Plato said:This is a busy-work problem.
Although I have not done the basic algebra, it appears that those two lines are skew lines. (you may need to show that)
Two skew lines share a unique perpendicular. Its length is the diameter of the sphere.
Now, one does not need to find that perpendicular. Just find the distance between the two lines.
Your text ought to discuss that somewhere. It may be in a problem set.