Using Orthogonal Projections in Linear Algebra

Question

Let $L$ be the line passing through the point $P_1 = (5, 3, −4)$ with direction vector $ d = (−3, −3, −1) $ and let $L_2$ be the line passing through the point $P_2 = (2, 1, 5)$ with the same direction vector. Find the shortest distance $d$ between these two lines.

Is it possible to use normal projection formula to find this answer? I seem to be missing a step... I would kill for some good tools or strategies to solve these efficiently.

Welcome to MSE! Take a look at MathJax basic tutorial and quick reference to learn how to properly typeset mathematical expressions in your posts. Also, these are parallel lines, are'nt? How would you do it in two dimensions? — mucciolo, Nov 18 '17 at 22:16
@Leigh: see here: https://math.stackexchange.com/questions/2213165/find-shortest-distance-between-lines-in-3d — Hector Blandin, Nov 18 '17 at 22:58
@Leigh: For parapell lines se here: https://math.stackexchange.com/questions/1347604/find-3d-distance-between-two-parallel-lines-in-simple-way — Hector Blandin, Nov 18 '17 at 23:00

score 0 · Answer 1 · answered Nov 18 '17 at 23:02

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If you know your lines are parallels you can use the formula of the distance between a point and a line which uses ortogonal projections. See David's answer in the bottom of the page here: Find 3D distance between two parallel lines in simple way

answered Nov 18 '17 at 23:02

Hector Blandin

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Using Orthogonal Projections in Linear Algebra

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