1. The problem statement, all variables and given/known data P1 = 3x+4y=1 P2 = x +y -z=3 Find the intersection. 2. Relevant equations x = x0 +at y = y0 + bt z = z0 + ct 3. The attempt at a solution I used a quicker method. I first did P1 x P2 and got its cross product <-4,3,-1> Then I followed what our teacher had shown us in class - we did substitution. 3x+4y=1 x+y -z =3 let x = 0, then 4y = 1 means y = 1/4 and y -z = 3 will give us -11/4 then i had both direction vector <-4,3,-1> and point (0,3,-11/4) and i could write the equations of the line of intersection using the parametric equations x = -4t + 0 y = 3t + 1/4 z = -1t -11/4 the book gives x = 4t +11 y = -3t - 8 z = t it seems like the author made z = 0 (in the parametric equation z0 = 0) my vector was <-4,3,-1> and his was <4,-3,1> and they were scalar multiple of -1 i know there are many possible parametric equations describing the same intersection. but is my result valid, even though my x0, y0, and z0 are different?