A vector perpendicular to a given vector 
is a vector 
(voiced "
-perp") such that 
and 
form a right angle.
In the plane, there are two vectors perpendicular to any given vector, one rotated 
counterclockwise and the other rotated 
clockwise. Hill (1994) defines 
to be the perpendicular vector obtained from an initial
vector
![a=[a_x; a_y]](/images/equations/PerpendicularVector/NumberedEquation1.svg) |
(1)
|
by a counterclockwise rotation by 
, i.e.,
![a^_|_=[0 -1; 1 0]a=[-a_y; a_x].](/images/equations/PerpendicularVector/NumberedEquation2.svg) |
(2)
|
In the plane, a vector perpendicular to 
can therefore be obtained by transposing the Cartesian
components and taking the minus sign of one. This operation is implemented in the
Wolfram Language as Cross[ax,
ay].
In three dimensions, there are an infinite number of vectors perpendicular to a given vector, all satisfying the equations
 |
(3)
|
