Question #267029

write the equation of the line that passes through the points (2,1,3) and (1,4,-3). write all three forms of the equations of the line.


1
Expert's answer
2021-11-17T10:05:49-0500

There is three major forms of linear equations:

  • vector form: (x,y,z)=(x0,y0,z0)+t(a,b,c)(x,y,z)=(x{\scriptscriptstyle 0},y{\scriptscriptstyle 0},z{\scriptscriptstyle 0})+t(a,b,c)
  • parametric form: x=x0+ta,y=y0+tb,z=z0+tcx=x{\scriptscriptstyle 0}+ta, y = y{\scriptscriptstyle 0}+tb,z=z{\scriptscriptstyle 0}+tc
  • symmetric form: xx0a=yy0b=zz0c{\frac {x-x{\scriptscriptstyle 0}} a}={\frac {y-y{\scriptscriptstyle 0}} b}={\frac {z-z{\scriptscriptstyle 0}} c}

Where x0,y0,z0x{\scriptscriptstyle 0}, y{\scriptscriptstyle 0},z{\scriptscriptstyle 0} - point on that line

(a, b, c) - directing vector of a line

t is a parameter, tRt \isin R

(a, b, c) = (1 - 2, 4 - 1, -3 -3) = (-1, 3, -6)

Then

  • vector form: (x,y,z)=(2,1,3)+t(1,3,6)(x,y,z)=(2,1,3)+t(-1,3,-6)
  • parametric form: x=2t,y=1+3t,z=36tx=2-t, y = 1+3t,z=3-6t
  • symmetric form: x21=y13=z36{\frac {x-2} {-1}}={\frac {y-1} 3}={\frac {z-3} {-6}}

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