Vector Problem: Finding Distance Between Point and Line L(t)

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SUMMARY

The discussion focuses on calculating the distance from a point P = (5, 0, -4) to the line defined by L(t) = <2-t, 1+4t, 4+2t>. The solution involves finding vectors connecting point P to the line L as functions of the parameter t. By calculating the squared lengths of these vectors, the goal is to determine the minimum value, which indicates the shortest distance from point P to line L. This approach emphasizes the use of calculus to find the minimum distance effectively.

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Homework Statement



Consider the line L(t) =<2-t,1+4t,4+2t> and the point P =(5,0,-4).
How far is P from the line L?

Homework Equations





The Attempt at a Solution


I'm confused on how to being this problem.

Any ideas would be great!
 
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Not a very geometric approach to this problem, but nevertheless gave a correct result, or at least the one that looks fine when plotted.

Anyway, find all the vectors connecting point P to the line L, all of them being a function of t (your parameter). This is pretty simple. Now, calculate the length of such vectors. Actually, length squared will do as well. This again is a function of t, isn't it? Now, what you want to find is the shorthest from all those vectors. So what do you do? Yep, you are looking for a minimum of the latter function. Once you find the value of the parameter, for which a vector LP is the shortest, you are at home :)
 

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