Finding a line orthogonal to another line Let $W$ be the line $\{(t,2t,-4t) : t \in \mathbb{R}\}$ Find an equation for $W^\perp$. 
I'm having a hard time figuring out how to find another line that is perpendicular to this one. The dot product must be zero, but since it's parameterized it throws me off. 
 A: I won't give you the whole answer, but I'd like to give you a hint.
Try to think of the slope of the line as $(1, 2, -4)$. What slope would be orthogonal to that vector? Once you have that, paramaterize the new line by multiplying it by $t$.
Good luck!
A: There are several ways to think about it, but using inner products is usually the most robust way to approach to geometric problems. It is generally believed that all geometric concepts can be expressed via inner products.
The space orthogonal to a straight line is a plane. There are several ways to describe a plane algebraically. Some are explicit, others are implicit. We'll go with an implicit one since it's the quickest (probably not best) way to answer this question.
You have a straight line that passes through the origin so we'll find a plane that passes through the origin. Every vector (x,y,z) in the plane is orthogonal to the vector $(1,2,-4)$ which points along the straight line. Expressed via the inner product, the orthogonality condition reads $$1x+2y-4z=0$$ and that's the implicit equation of the plane!
