Question

Determine vector, parametric, and if possible, symmetric equations of the line through A(-3, 5, -5) and B(9, 2, -1).

Answer 1

direction vector AB = <12, -3, 4>

so a possible vector equation is
<x,y,z> = <-3,5,-5> + t<12,-3,4>

from that, it is a piece of cake to form the other two versions.
Let me know what you get.

Answer 2

{ vector } = < -3 + 12t, 5 - 3t , -5 + 4t > { parametric } ====> [ x + 3 ] / 12 = [ y - 5 ] / - 3 = [ z + 5 ] / 4 {= t ; symmetric }

Answer 3

for parametric, the more common form would be:
x = -3 + 12t
y = 5 - 3t
z = -5 + 4t

your symmetric version is correc

Answer 4

could i also write the symmetric equation like this?
(x + 3) / 12 = (y - 5) / -3 = (z + 5) / 4

Answer 5

Isn't that what you first had in your original post ?

Another version would use the other point (9, 2, -1)
It would be :
(x-9)/12 = (y-2)/-3 = (z+1)/4