Exercise 6

Suppose that $$\vvec_1=\threevec103,\hspace{24pt} \vvec_2=\threevec222$$ form the basis for a plane $W$ in $\real^3$.

1. Find a basis for the line that is the orthogonal complement $W^\perp$.

2. Given the vector $\bvec=\threevec6{-6}2$, find $\yvec$, the orthogonal projection of $\bvec$ onto the line $W^\perp$.

3. Explain why the vector $\zvec = \bvec-\yvec$ must be in $W$ and write $\zvec$ as a linear combination of $\vvec_1$ and $\vvec_2$.