Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 1

Expanding the cross product componentwise givesIts dot product with each original vector is zero, as required.

First $(1,-1,0{)}^T\times (0,0,1{)}^T=(-1,-1,0{)}^T$. Taking the remaining cross product givesThe order of the two cross products matters because the operation is not associative.