Exercise 6

Consider the vectors $$\vvec_1=\threevec204,\hspace{24pt} \vvec_2=\threevec{-1}2{-4}.$$ and a general vector $\xvec=\threevec xyz$.

1. Write an equation in terms of $x$, $y$, and $z$ that describes all the vectors $\xvec$ orthogonal to $\vvec_1$.

2. Write a linear system that describes all the vectors $\xvec$ orthogonal to both $\vvec_1$ and $\vvec_2$.

3. Write the solution set to this linear system in parametric form. What type of geometric object does this solution set represent? Indicate with a rough sketch why this makes sense.

4. Give a parametric description of all vectors orthogonal to $\vvec_1$. What type of geometric object does this represent? Indicate with a rough sketch why this makes sense.