Planes and the Cross Product

Here are the questions we considered in today’s class. See if you can answer these (with algebraic solutions) for tomorrow’s lesson. GeoGebra will be useful to check you answers, and to give you some insight into the question if you get stuck.

1. Find a vector normal to both
   \[\vec{a}=\begin{bmatrix}1\\2\\-2\end{bmatrix} \text{ and }\vec{b}=\begin{bmatrix}3\\4\\1\end{bmatrix}.\]
2. a) Find a vector normal to the plane \[\vec{r}=\begin{bmatrix}1\\4\\-2\end{bmatrix} +\lambda \begin{bmatrix}1\\1\\-1\end{bmatrix} + \mu \begin{bmatrix}-3\\1\\2\end{bmatrix}\] b) Using your answer to part a), can you find the distance of the point \(A(1, 1, 1)\) to the given plane?