Write a unit vector perpendicular to both the vectors a = ^i + ^j + ^k and b = ^i + ^j.

A unit vector in the direction a × b is perpendicular to both the vectors. So, a × b = (^i + ^j + ^k) × (^i + ^j) = ||^i ^j ^k||1 1 1||1 1 0|| = −^i + ^j and |a × b| = √1² + 1² = √2. Thus, the required vector is ±(−^i + ^j)/√2.