What is the z component of a vector parallel to the x-y plane ? explain
find all unit vectors parallel to the x-y plane and perpendicular to the vector [1,-2, 2 ]
In $\displaystyle \mathbb{R}^3$ any vector in the xy plane has no vertical component so its z component is zero.
for part two
is the vector is parallel to the xy plane it is perpendicular to the z axis
again let <x,y,z> be any vector so we know that
$\displaystyle <0,0,1> \cdot <x,y,z>=0 \iff z=0$
$\displaystyle <x,y,0> \cdot <1,-2,2>=0 \iff x-2y=0 \iff x=2y$
so we get
<x,2x,0> =x<1,2,0>
To make this a unit vector divide each component of the vector by the vectors magnitude!