# Vector geometry - Intersection of lines

• Keshroom

## Homework Statement

I have 2 parametric vector equations (of a line)

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

how do i find the coordinates for which they intersect each other?

## Homework Equations

x=a+λv, for some λ in ℝ (parametric vector form of line)

## The Attempt at a Solution

As in high school, with the form y=mx+b i would make the 2 equations equal to each other, solve for x, then substitute back into either equations to find y.

I've tried making the (x,y,z) components equal to each other, solve for 't' and substitute back in but i can't get the answer in the back of the book

parametic equations
for r(t): x = 2+t, y= -4-3t, z= 4+4t
for s(t): x = 1+2t, y= -1-t, z= t

Now i did
2+t = 1+2t
t = 1

substituting back into x=2+t: x = 3

i did this for also y and x components and got (3, 1/2, -4/3)
hmmmm
I have a feeling that this method isn't correct :s

Change the notation to
r(t) = (2,-4,4) + t(1,-3,4)
s(u) = (1,-1,0) + u(2,-1,1)
the parameters t and u don't have to be the same for the lines to intersect each other.

Change the notation to
r(t) = (2,-4,4) + t(1,-3,4)
s(u) = (1,-1,0) + u(2,-1,1)
the parameters t and u don't have to be the same for the lines to intersect each other.

alright. Now how do i solve it?

alright. Now how do i solve it?

The same way you tried before. Equate components of the vectors and solve them. Try it. The first component gives you 2+t=1+2u. Solve that for t and substitute into the rest.