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it is possible to define a 3d vector using the form:

(x,y,z) = (x_1,y_1,z_1)+t(a,b,c)

this can then be parameterized to become

x = x_1 + a*t

y = y_1 + b*t

z = z_1 +c*t

so, for example, given the vector (x,y,z) = (1,2,3) + t(5,6,7)

one can parametrize it to become:

x = 1 + 5t

y = 2 + 6t

z = 3 + 7t

solving the first equation for t gives

t = (x-1)/5

plugging that into the other two equations gives:

y = 2 + 6/5 * (x-1)

z = 3 + 7/5 * (x-1)

solving the second one for x gives:

x = (5*z-8)/7

changing y = 2 + 6/5 * (x-1) to y = 2 + 6/5 * (2x-x-1)

and in the -x spot putting -(5*z-8)/7 yeilds:

y = 2 + 6/5 * (2x-(5*z-8)/7-1)

simplifying gets:

76 = 35 y + 30 z - 84 x

which is the equation of a plane.

this cannot possibly be right as i started out with a 2-d object (the vector)

where did i mess up?

(p.s. i tried to do all of this algebraically but it got too ugly for me)

thanks.