# Find equation of a plane containing two lines

## Homework Statement

find the equation of the plane that contains the line r_1(t) = (t,2t,3t) and r_2(t)=(3t,t,8t)

## The Attempt at a Solution

i don't know where to start...my book does not have an example similar. can somebody just point me to the right direction?

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Mark44
Mentor
Each of your line equations defines a vector with the same direction as the line. Find the cross product of these vectors to get a third vector, say <a, b, c>. That will be a normal to the plane. Find a point on either of your given lines, say (x0, y0, z0).

Use the normal and the given point to write the equation of the plane as a(x - x0) + b(y - y0) + c(z - z0) = 0.

$$r_1=t(1,2,3)$$

$$r_2=t(3,1,8)$$

Both $$\vec{v}_1=\left(\begin{array}{c} 1 \\ 2 \\ 3\end{array}\right)$$ and $$\vec{v}_2=\left(\begin{array}{c} 3 \\ 1 \\ 8\end{array}\right)$$ are in the plane.

So $$\vec{n}=\vec{v}_1 \times \vec{v}_2=\left(\begin{array}{c} 13 \\ 1 \\ -5\end{array}\right)$$ is normal to the plane.

Hence the equation:

$$13x+y-5z=0$$

Mark44
Mentor
Donaldos,
It is the policy of this forum to provide help to a poster, but not to give a complete answer to someone's problem.

ok i got the answer, but when i did the cross product my answers sitll have the t in them

my ans is 13t2x+t2y-5t2z = 0

the books answer is exactly that but without the t's

Donaldos,
It is the policy of this forum to provide help to a poster, but not to give a complete answer to someone's problem.
I'm sorry. I'll keep that in mind.

i see donaldos wrote r with the t outside before doing the determinant...what happens to that?

Mark44
Mentor
All you need is any old vector that is parallel to the line, so any multiple of the vector will still be parallel. The t multiplier can be any real value, so it's convenient to let t = 1.