Question

Let L1 be the line passing through the points Q1=(1, −1, −1) and Q2=(−3, 5, −5) and let L2 be the line passing through the point P1=(1, −5, 2) with direction vector →d=[−2, −1, 1]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

Answer 1

Express the lines in parametric form,
set the components equal to each other and solve the parameter values

Answer 2

you could also define the plane that would contain the lines, and see if the other lines anchor point is in the same plane

Answer 3

how do i make them both in parametric form?

Answer 4

do you know how to write their equations in vector form ?

Answer 5

yeah i think so

Answer 6

can i use a matrix to solve for the parameters?

Answer 7

sure, you can

Answer 8

is there an easier way?

Answer 9

will x,y,z all be equal on both lines?

Answer 10

project them onto the xy plane and see where they cross ... then determine if they have the same z value for that points

not sure if it 'easier'

Answer 11

solving system of 3 equations shouldn't be hard i guess ?

Answer 12

it is painful, but since you're doing linear algebra you should be good at solving system of linear equations :)

Answer 13

q1=(1, −1)
q2=(−3, 5)
----------
4,-6, slope of -3/2

p1=(1, −5) with direction vector [−2, −1], slope of 1/2

Answer 14

fml thanks you are really helpful

Answer 15

amsitre64 i'm not really understanding the "planes" way of solving it

Answer 16

so are you able to figure out how to express the given line in vector form yet ?

Answer 17

*lines

Answer 18

cross the direction vectors, anchor it to one of the points, and see if the other points fit in the plane .... lines only cross if they can fit in the same plane

Answer 19

i think i can yeah

Answer 20

if all the points given are in the same plane ... then the lines cross but its not really going to tell us where they cross at