# Thread: Lines and planes question

1. ## Lines and planes question

If a plane is defined by two lines, and I have one line, like this

r=(a,b,c) + t (d,e,f)

Then can I just arbitrarily add other direction vectors and all of those new planes must intersect at that line?

r=(a,b,c) + t (d,e,f) + s(5,4,7)
r=(a,b,c) + t (d,e,f) + s(9,1,3)

etc...

2. Originally Posted by theowne
If a plane is defined by two lines, and I have one line, like this r=(a,b,c) + t (d,e,f)
Then can I just arbitrarily add other direction vectors and all of those new planes must intersect at that line?
r=(a,b,c) + t (d,e,f) + s(5,4,7)
r=(a,b,c) + t (d,e,f) + s(9,1,3)
This appears to be a continuation of a previous question. I was reluctant to try to answer because the way you posted it shows some real confusion in concepts.
First, $\displaystyle l(t) = \left\langle {a,b,c} \right\rangle + t\left\langle {d,e,f} \right\rangle$ is a line.
Second, $\displaystyle \pi _1 (t,s) = \left\langle {a,b,c} \right\rangle + t\left\langle {d,e,f} \right\rangle + s\left\langle {h,j,k} \right\rangle$ is a plane provided that the vectors $\displaystyle \left\langle {d,e,f} \right\rangle \,\& \,\left\langle {h,j,k} \right\rangle$ are independent, (i.e. they are not parallel, that is they are not multiples of each other).
Note that if $\displaystyle s=0$ then $\displaystyle l(t) \subseteq \pi _1 (t,s)$.
Thus, you are correct both planes $\displaystyle \pi _1 (t,s) = \left\langle {a,b,c} \right\rangle + t\left\langle {d,e,f} \right\rangle + s\left\langle {5,4,7} \right\rangle \,\& \,\pi _2 (t,s) = \left\langle {a,b,c} \right\rangle + t\left\langle {d,e,f} \right\rangle + s\left\langle {9,1,3} \right\rangle$ contain the line $\displaystyle l(t)$.
But note that you must pick vectors that are independent of the direction vector.

3. Well, let's say the you're given a line,
r=(1,2,3) + t(4,5,6)

If you have another direction vector, you can define a plane, right?

eg

=(1,2,3) + t(4,5,6) + s(5,4,7)

This is a plane which passes through the above line, right?

How about if arbitrarily replace the above s vector with another, like

=(1,2,3) + t(4,5,6) + s(9,1,3)

These both pass through the mentioned line, then? So both of these planes intersect at that line? Is that all there is to it?