# Cartesian Equation of a Plane

#### coma

##### New member
"Write the cartesian equation of the plane containing the line 𝑟⃗1 = (1, 2, 4) + 𝑡(4, 1, 11) and perpendicular to 𝑟⃗2 = (4, 15, 8) + 𝑠(2, 3, −1)"

Haven't been able to figure out where to even begin with this one, visualizing it in my head makes sense but I've got no idea how to solve it algebraically. It's worth 3 marks if that's helpful.

#### Dr.Peterson

##### Elite Member
"Write the cartesian equation of the plane containing the line 𝑟⃗1 = (1, 2, 4) + 𝑡(4, 1, 11) and perpendicular to 𝑟⃗2 = (4, 15, 8) + 𝑠(2, 3, −1)"

Haven't been able to figure out where to even begin with this one, visualizing it in my head makes sense but I've got no idea how to solve it algebraically. It's worth 3 marks if that's helpful.
What forms have you learned for the equation of a plane?

You have a normal vector, and a lot of points in the plane (a whole line's worth), so you should have an appropriate form for the equation. (In fact, you could have too much information -- you'll need to check that the first line actually lies in this plane. That may be what is confusing you.)

#### pka

##### Elite Member
"Write the cartesian equation of the plane containing the line 𝑟⃗1 = (1, 2, 4) + 𝑡(4, 1, 11) and perpendicular to 𝑟⃗2 = (4, 15, 8) + 𝑠(2, 3, −1)"
Let's rewrite the notation.
$${\ell _1}(t) = \left\{ \begin{gathered} x = 1 + 4t \hfill \\ y = 2 + t \hfill \\ z = 4 + 11t \hfill \\ \end{gathered} \right.~~$$ $${\ell _2}(s) = \left\{ \begin{gathered} x = 4 + 2s \hfill \\ y = 15 + 3s \hfill \\ z = 8-s \hfill \\ \end{gathered} \right.$$
Now we need the point $$(1,2,4)$$ and the normal vector $$\vec{n}=\left<2,3,-1\right>$$

#### pka

##### Elite Member
Let's rewrite the notation.
$${\ell _1}(t) = \left\{ \begin{gathered} x = 1 + 4t \hfill \\ y = 2 + t \hfill \\ z = 4 + 11t \hfill \\ \end{gathered} \right.~~$$ $${\ell _2}(s) = \left\{ \begin{gathered} x = 4 + 2s \hfill \\ y = 15 + 3s \hfill \\ z = 8-s \hfill \\ \end{gathered} \right.$$
Now we need the point $$(1,2,4)$$ and the normal vector $$\vec{n}=\left<2,3,-1\right>$$
Edit P. S. correction of a miss-copy.
$$\vec{n}=\left<4,1,11\right>\times\left<2,3,-1\right>=\left<-17,3,5\right>$$

#### pka

##### Elite Member
Edit P. S. correction of a miss-copy.
$$\vec{n}=\left<4,1,11\right>\times\left<2,3,-1\right>=\left<-17,3,5\right>$$
A second correction.
$$\vec{n}=\left<4,1,11\right>\times\left<2,3,-1\right>=\left<-17,13,5\right>$$

@pka