# Intersection of a plane and a line in vector form.

I'm stuck with this question. Consider the plane $$E$$ with cartesian equation $$x - 2y + az = 6$$, and the line $$L$$ with vector form $$(x, y, z) = (1, -1, 0) + t(-3, -a, a)$$. For what values of $$a$$, if any do $$L$$ and $$E$$ intersect?

I find it hard to visualise what these 2 equations look like and what their intersection would be like. I'm also not really sure where to start with these questions in general. Thanks.

New contributor
Fireflies Fire Fly is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.

You have a line given with a point $$\vec p$$ and a direction $$\vec r$$ as

$$L=\{\vec p+t\vec r\in\Bbb R^3 \mid t\in\Bbb R\} \tag 1$$

where $$\vec r$$ is a function of $$a$$.

The plane $$E$$ is given by a vector $$\vec n$$ that's normal to $$E$$. Given a point $$\vec q\in E$$, all points $$\vec v\in E$$ satisfy $$(\vec v-\vec q)\cdot \vec n = 0\tag2$$ because the line between any two points in the plane is perpendicular to the normal $$\vec n$$. Now $$\vec q\cdot\vec n = d$$ is just some real number that's related to the distance between $$E$$ and the origin. Hence, $$E$$ can be expressed as

$$E = \{\vec v\in\Bbb R^3 \mid \vec v\cdot\vec n=d\}\tag 3$$

If we intersect $$L$$ with $$E$$, then all points in $$L\cap E$$ must satisty the contition in $$(3)$$, so insert the term $$(1)$$ that represents points in $$L$$ into $$(3)$$:

$$L\cap E = \{ \vec p+t\vec r \in\Bbb R^3 \mid t\in\Bbb R, (\vec p+t\vec r)\vec n = d\} \tag 4$$

This gives a determinig equation for $$t$$:

$$t = \frac{d-\vec p\cdot\vec n}{\vec r\cdot \vec n}\tag 5$$

Now you have to drop in values and determine for which values of $$a$$ there are solutions, and how many solutions there are. In your specific case:

\begin{align} d &= 6 \\ \vec p &= (1,-1,0) \\ \vec r &= (-3,-a,a) \\ \vec n &= \cdots \end{align}

As your $$\vec n$$ also depends on $$a$$, you'll get an equation in $$a$$ that's quadratic or linear (depending on how you resolve that typo in the specification of $$E$$).