Find Line Intersection: Symmetric Equations

[Parth Dave]

Find symmetric equations of the line that passes through the point (0,1,2) and meets each of the lines x = y = z + 2 and x/-2 = (y+3)/1 = z/3.

Those equations can be written as:
r = (0, 0, 2) + t(1, 1, 1)
r = (0, -3, 0) + s(-2, 1, 3)


Now, I can't seem to find any direction to go with this. I tried a whole lot of different things that all eventually led nowhere. First, I gave co-ordinates to the intersection points and then i created to slopes in between these points. But i eventually came up with equations with like 8 variables in them so I couldn't figure that one out. Can someone lead me in some sort of direction? Any would would be appreciated.

 

[Parth Dave]

Btw, i do have the solution,

x/2 = (y-1)/-1 = (z-2)/-2

if anyone is curious.

 

[wais]

One possible approach to finding the intersection point of these two lines is to set up a system of equations using the symmetric equations. Since the point (0,1,2) lies on the line with equation r = (0, 0, 2) + t(1, 1, 1), we can substitute these values into the equation for the line and solve for t. This will give us the value of t at the point of intersection.

Similarly, we can substitute the coordinates of the point (0,1,2) into the equation for the second line and solve for s. This will give us the value of s at the point of intersection. Once we have both t and s, we can substitute these values back into the equations for the lines to find the coordinates of the intersection point.

Another approach could be to use vector equations to find the intersection point. We can represent each line as a vector equation and then set them equal to each other. This will give us a system of equations that we can solve for the coordinates of the intersection point.

In either case, it may be helpful to draw a diagram or visualize the lines in 3D space to get a better understanding of their relationship and how they intersect. Keep in mind that there may be multiple points of intersection or no intersection at all, so it is important to carefully check your solutions.