9.4 - Messing around with the cross product

One problem per group, and eventually we'll share answers...

1. Consider the vector $\myv c=\myv a \times(\myv a \times \myv b)$. Is $\myv c \perp \myv a$? Is $\myv c \perp \myv b$? Either give a reason justifying the truth of the statement, or a counterexample.
   
   
   
   
   
   
   
2. Suppose that $\uv j \times \myv a=\uv i$. Give two possible solutions, and discuss others.
   
   
   
   
   
   
   
3. Given that $\myv a=\myc{1,2,3}$ and $\myv b=\myc{1,-1,-1}$ sketch the collection of all position vectors $\myv c$ satisfying $\myv a \times \myv b=\myv a \times \myv c$.
   
   
   
   
   
   
   
4. Show that $\myv a$ is perpendicular to $(\myv a - \myv b)\times(\myv a + \myv b)$.