N2009 P2 Q2

Relative to the origin \(O\), two points \(A\) and \(B\) have position vectors given by \(\mathbf a=14\mathbf i+14\mathbf j+14\mathbf k\) and \(\mathbf b=11\mathbf i-13\mathbf j+2\mathbf k\) respectively.

1. The point \(P\) divides the line \(AB\) in the ratio \(2:1\). Find the coordinates of \(P\).[2]
2. Show that \(AB\) and \(OP\) are perpendicular.[2]
3. The vector \(\mathbf c\) is a unit vector in the direction of \(\overrightarrow{OP}\). Write \(\mathbf c\) as a column vector, and give the geometrical meaning of \(|\mathbf a\cdot\mathbf c|\).[2]
4. Find \(\mathbf a\times\mathbf p\), where \(\mathbf p\) is the vector \(\overrightarrow{OP}\), and give the geometrical meaning of \(|\mathbf a\times\mathbf p|\). Hence write down the area of triangle \(OAP\).[4]