How to get a horizontal normal vector from a vector

Question

I have a vector in 3d. Z is the vertical axis. I would like to get those 2 vectors which are perpendicular to the original vector and are horizontal (Z=0). Its length does not matter to me. Thank you for your help!

Answer 1

If the $z$ component of a vector is zero then the vector is orthogonal to the $z$ axis.

So you want a vector orthogonal to the $z$ axis and also orthogonal to one other vector you have received.

The usual way to find a vector orthogonal to two other vectors in three dimensions is to take the cross product of the two vectors. That is, you take your given vector $V,$ set $U = (0,0,1)$ (parallel to the $z$ axis) and take $$ U \times V .$$

But since one of your two vectors is parallel to the $z$ axis, in your case there is a simpler method. You can project your given vector $V$ onto the $x,y$ plane simply by replacing its $z$ component with $0$. For example, if you were given $V=(4,5,6),$ project it to $(4,5,0).$ Now rotate the projected vector by $90$ degrees in the $x,y$ plane.

Once you have one vector orthogonal to your given vector and to the $z$ axis, any other non-zero multiple of that vector--including negative multiples, which "point" in the opposite direction--is also orthogonal to your given vector and to the $z$ axis.

Note that neither of these methods works correctly when the given vector $V$ is parallel to the $z$ axis. But then any vector parallel to the $x,y$ plane (any vector with zero as its $z$ coordinate) will be a solution.