N2010 P1 Q1

The position vectors \(\mathbf a\) and \(\mathbf b\) are given by \(\mathbf a=2p\mathbf i+3p\mathbf j+6p\mathbf k\) and \(\mathbf b=\mathbf i-2\mathbf j+2\mathbf k\), where \(p>0\). It is given that \(|\mathbf a|=|\mathbf b|\).

1. Find the exact value of \(p\).[2]
2. Show that \((\mathbf a+\mathbf b)\mathbin{\cdot}(\mathbf a-\mathbf b)=0\).[3]