Question
If Lsub1 has parametric equation x=1+3t, y=1+t, z=4-t, and Lsub2 has parametric equation x=7-6t, y=-2t, z=3+2t, then Lsub1 and Lsub2 are parallel.
The answer is true.
Please show me why it is true. Thank you very much.
Hello Jenny,
your equations describe two straight lines in
By comparison you can see, that
. That means the direction vectors are collinear: They have the same direction but different length.
Therefore and are at least parallel. To proof if they are actually the same you have to show that the fixed point of belongs to too.
EB
Hello, Jenny!
If has parametric equations: .
and has parametric equations: .
then and are parallel.
Two lines are parallel if their direction vectors are parallel.
. . (They do not have to be collinear.)
has direction vector:
has direction vector:
Since . . . . Q.E.D.
Two lines are parallel if their direction vectors are parallel.
. . (They do not have to be collinear.)
L_1 has direction vector: \vec{u}\:=\:\langle 3,1,\text{-}1\rangle
L_2 has direction vector: \vec{v}\:=\:\langle\text{-}6,\text{-}2,2\rangle \:=\:\text{-}2\langle3,1,\text{-}1\rangle
Since \vec{v} = -2\vec{u}\!:\;\;\vec{u} \parallel \vec{v} . . . . Q.E.D.
Hi Soroban,
Thank you very much!