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.

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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.

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What forms have you learned for the equation of a plane?

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.

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

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Let's rewrite the notation."Write the cartesian equation of the plane containing the line πβ1 = (1, 2, 4) + π‘(4, 1, 11) and perpendicular to πβ2 = (4, 15, 8) + π (2, 3, β1)"

\({\ell _1}(t) = \left\{ \begin{gathered} x = 1 + 4t \hfill \\ y = 2 + t \hfill \\ z = 4 + 11t \hfill \\ \end{gathered} \right.~~\) \({\ell _2}(s) = \left\{ \begin{gathered} x = 4 + 2s \hfill \\ y = 15 + 3s \hfill \\ z = 8-s \hfill \\ \end{gathered} \right.\)

Now we need the point \((1,2,4)\) and the normal vector \(\vec{n}=\left<2,3,-1\right>\)

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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 \(\vec{n}=\left<2,3,-1\right>\)

\(\vec{n}=\left<4,1,11\right>\times\left<2,3,-1\right>=\left<-17,3,5\right>\)

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A second correction.Edit P. S. correction of a miss-copy.

\(\vec{n}=\left<4,1,11\right>\times\left<2,3,-1\right>=\left<-17,3,5\right>\)

\(\vec{n}=\left<4,1,11\right>\times\left<2,3,-1\right>=\left<-17,13,5\right>\)

Your original post is correct.

@coma

You can write down the equation of a plane if you know a point in the plane and a vector normal to the plane. You can read off a point from the first equation and read off a normal vector from the direction vector in the second equation, as indicated by @pka in his original post.