Find the distance between a point and a line (given its vector equation)
1. The problem statement, all variables and given/known data
Let [tex]l_1:<x,y,z>=<2,1,3>+t<1,2,1>[/tex] and [tex]P(1,3,2)[/tex] be a line and point in R^{3}, respectively. Find the distance from P to l. 2. Relevant equations distance between two points in R^{3} [tex]d=\sqrt{(xx_0)^2+(yy_0)^2+(zz_0)^2}[/tex] and the line in this problem is given to us in vector equation form, from which I can find the directional vector v and the position vector r_{0} 3. The attempt at a solution I'll need a bit of guidance on this problem. I believe that I'm supposed to find the shortest distance between P and some point on line L, and I can only think that the shortest distance between P and some point on L would be some path from P that intersects L at a 90 degree angle (perpendicular to L). I was thinking of perhaps using L's directional vector and the point P in order to construct some line that goes through P and is perpendicular to L, which would mean I would need to cross the directional vector with something? I'm not too sure on how to go about this, or whether or not this is a proper way of approaching this problem. Any guidance would be greatly appreciated. 
Re: Find the distance between a point and a line (given its vector equation)
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Re: Find the distance between a point and a line (given its vector equation)
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Now if take the cross product PQ x U and look at its maginitude: PQ x U = PQUsin(θ) = PQsin(θ) = d. To summarize: To get the distance from a point to a line take the magnitude of the cross product of a vector from any point on the line to the external point with the unit direction vector. 
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