March 27, 2017

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this can be done by projections of two vector.
recall that the scalar projection of vector b on vector a is a∙b/│a│

so let's find a point on the give line,
e.g. the point B(6,0,1) (I let t=0)

draw a perpendicular from your given point P(1,-5,2) to meet the line at Q
So PQB forms a righ-angled triangle with │vector BP│ the hypotenuse.

vector BP, lets call it vector b, is (5,5,-1) and vector a is (3,1,2), (from your given line)

so the projection of vector b on vector a
= (5,5,-1)∙(3,1,2)/│(3,1,2)│
= (15+5-2)/√(9+1+4)
= 18/√14

also │b│ = √(25+25+1) = √51

You now have 2 sides of a right-angled triangle, with your required distance as the third side.
I will let you finish the arithmetic.
What do you with the triangle? How do you know which side is which, are you finding the hypotenuse, or just a side? This changes the math done. Like a^2+b^2=c^2, but if you are finding a side then it would be c^2-a^2=b^ can you please explain, thanks.

  • Calc. for Reiny - ,

    Since you posted my reply, I will use my letters

    I found │vector BP│ to be √51 , the hypotenuse
    I found │vector bQ│ to be 18/√14
    You want │vector PQ│

    so (PQ)^2 + (BQ)^2 = (BP)^2
    (PQ)^2 = (BP)^2 - (BQ)^2
    = 51 - 324/14
    = 390/14
    BP = √(390/14)
    = 5.278

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