Question

Determine the vector equation of a plane that contains the line with symmetric equation (x+3)/-2=(y+1)/5=(z+2)/4 and the point P(1,3,0).

For this question to find the direction vector you would use subtraction with the points (-3,-1,-2) and (1,3,0). Why would it be (1,3,0)-(-3,-1,-2) instead of (-3,-1,-2)-(1,3,0)? is it because you need to subtract the smaller number from the larger one?

Answer 1

A point on the line is (-3,-1,-2) and the direction of the line is [-2,5,4]

To find the equation of a plane I need another direction vector,
using the point (-3,-1,-2) and the other point P(1,3,0) , I get [-4, -4, -2] or we could simply use
[2,2,1]

since you want the vector equation, we can now just state that using parameters s and t

r = (1,3,0) + s[-2,5,4] + t[2,2,1]

of course you know that this equation is not unique.

your question about the direction of the subtraction is a mute one, of course it makes no difference, we could have used the vector
(4,4,2) or (-4,-4,-2) or (-2,-2,-1) or (8000,8000,1000)