Intersection of Lines (Vectors)

[FeDeX_LaTeX]

Homework Statement

Show that if the two lines

##\frac{x - c_1}{d_1} = \frac{y - c_2}{d_2} = \frac{z - c_3}{d_3}## and

##\frac{x - d_1}{c_1} = \frac{x - d_2}{c_2} = \frac{x - d_3}{c_3}##

intersect, they lie in the plane ##r.(c \times d) = 0##

where c = c₁i + c₂j + c₃k and d = d₁i + d₂j + d₃k

The Attempt at a Solution

I've shown that they intersect at ##x = c_1 + d_1, y = c_2 + d_2, z = c_3 + d_3##. It seems intuitively obvious that the lines lie in the plane r.(c x d) = 0, but I just can't seem to properly show this. Any advice?

 

[verty]

It is possible for the lines to intersect at infinitely many points, for example let ##c_i = d_i = i##. In this case, ##r . (c × d) = 0## isn't a plane. So I think one must assume we are talking about lines that intersect at exactly one point, if at all.

Given this, and assuming your proof is correct that they intersect at c+d if they intersect at all, showing that the points (c, d, c+d) are in that plane will suffice because each line goes through two of those points.

 

[FeDeX_LaTeX]

Thanks -- I assume that the lines are non-parallel. All three points lie in the plane.

 

[haruspex]

The question is a little odd. Those two lines necessarily intersect (at x = c+d).
Given an equation like ##\frac{x - c_1}{d_1} = \frac{y - c_2}{d_2} = \frac{z - c_3}{d_3}##, you can set each term to equal some scalar parameter t, then express the line as a vector equation. It should then be obvious that the line lies in the subspace (be it a line or a plane) generated by c and d.