# Here Are 4 Ways to Find the Distance Between a Point and a Line in 3D

Working with stuff in 3D can seem complicated — but if you embrace the idea of vectors, everything can work out just fine.

One of the problems you will see in your calculus course (usually in Calc-3) is to find the shortest distance between a point and a line (in 3D). I’ve already gone over the idea of parametric equations and the intersection between lines: check it out.

As a physicist (that’s me), I like to think of a 3D line like it’s the trajectory of a ball moving with a constant velocity. If a ball has a velocity (vector v) and starts at a vector position r0 at time t = 0, then the positions for all future times can be described with the following equation:

This is the same as the three component equations:

Boom. There’s your parametric equations for a 3D line. Pick a value for t and you can get your x, y, z values…

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Physics faculty, science blogger of all things geek. Technical Consultant for CBS MacGyver and MythBusters. WIRED blogger.